Intersection testing in a ray tracing system using a ray coordinate system

ABSTRACT

A method and an intersection testing module for performing intersection testing of a ray with a convex polygon in a ray tracing system. The ray and the convex polygon are defined in a 3D space using a space-coordinate system. The ray is defined with a ray origin and a ray direction. A ray-coordinate system is used to perform intersection testing, wherein the ray-coordinate system has an origin at the ray origin, and wherein the ray-coordinate system has three basis vectors. A first of the basis vectors is aligned with the ray direction. A second and a third of the basis vectors: (i) are both orthogonal to the first basis vector, (ii) are not parallel with each other, and (iii) have a zero as one component when expressed in the space-coordinate system. A result of performing the intersection testing is outputted for use by the ray tracing system.

FIELD

The present disclosure is directed to techniques of performingintersection testing in a ray tracing system.

BACKGROUND

Ray tracing is a computational rendering technique for generating animage of a scene (e.g. a 3D scene) by tracing paths of light (‘rays’)usually from the viewpoint of a camera through the scene. Each ray ismodelled as originating from the camera and passing through a pixel intothe scene. As a ray traverses the scene it may intersect objects withinthe scene. The interaction between a ray and an object it intersects canbe modelled to create realistic visual effects. For example, in responseto determining an intersection of a ray with an object, a shader program(i.e. a portion of computer code) may be executed in respect of theintersection. A programmer can write the shader program to define howthe system reacts to the intersection which may, for example cause oneor more secondary rays to be emitted into the scene, e.g. to represent areflection of the ray off the intersected object or a refraction of theray through the object (e.g. if the object is transparent ortranslucent). As another example, the shader program could cause one ormore rays to be emitted into the scene for the purposes of determiningwhether the object is in shadow at the intersection point. The result ofexecuting the shader program (and processing the relevant secondaryrays) can be the calculation of a colour value for the pixel the raypassed through.

Rendering an image of a scene using ray tracing may involve performingmany intersection tests, e.g. billions of intersection tests forrendering an image of a scene. In order to reduce the number ofintersection tests that need to be performed, ray tracing systems cangenerate acceleration structures, wherein each node of an accelerationstructure represents a region within the scene. Acceleration structuresare often hierarchical (e.g. having a tree structure) such that theyinclude multiple levels of nodes, wherein nodes near the top of theacceleration structure represent relatively large regions in the scene(e.g. the root node may represent the whole scene), and nodes near thebottom of the acceleration structure represent relatively small regionsin the scene. A “tree node” refers to a node which has pointers to othernodes in the hierarchical acceleration structure, i.e. a tree node haschild nodes in the hierarchical acceleration structure. A “leaf node”refers to a node which has one or more pointers to one or moreprimitives, i.e. a leaf node does not have child nodes in thehierarchical acceleration structure. In other words, leaf nodes of theacceleration structure represent regions bounding one or more primitivesin the scene. The acceleration structure can have different structuresin different examples, e.g. a grid structure, an octree structure, aspace partitioning structure (e.g. a k-d tree) or a bounding volumehierarchy. The nodes can represent suitable shapes or regions in thescene (which may be referred to herein as “boxes”). In some examples thenodes represent axis-aligned bounding boxes (AABBs) in the scene.

Intersection testing can be performed for a ray (e.g. in a recursivemanner) using the acceleration structure by first testing the ray forintersection with the root node of the acceleration structure. If theray is found to intersect a parent node (e.g. the root node), testingcan then proceed to the child nodes of that parent. In contrast, if theray is found not to intersect a parent node, intersection testing of thechild nodes of that parent node can be avoided, saving computationaleffort. If a ray is found to intersect a leaf node then it can be testedagainst the objects within the region represented by the leaf node tothereby determine which object(s) the ray intersects with. The objectsmay be represented as convex polygons. Often the convex polygons aretriangles, but they may be other shapes, e.g. squares, rectangles,pentagons, hexagons, etc. If more than one intersection is found for aray then the closest of the intersection points to the ray's origin(i.e. the first intersection that the ray encounters in the scene) maybe identified and the ray may be determined to intersect at thisidentified closest intersection. It is possible that there may bemultiple closest hits for a ray, and in this case some tie-break logicmay be used to select one of the multiple closest hits to use as theidentified closest intersection. For some types of rays, the closestintersection might not need to be identified. For example, whenprocessing shadow rays, an indication that there is at least oneintersection is sufficient, without determining which of theintersections is the closest, and some APIs may allow the traversal ofan acceleration structure for shadow rays to be terminated in responseto finding any intersection, to thereby reduce the number ofintersection tests that need to be performed.

A ray (r) can be defined as r=O+Dt where O is a vector which representsthe ray origin, D is a vector which represents the ray direction and trepresents a distance along the ray from the origin. A primitive can berepresented as a convex, or further strictly convex, polygon. A polygon(e.g. triangle) is defined by an ordered set of planar vertices wherebyconsecutive pairs of vertices define the polygon edges and the overallvertex order (its winding order) is used to indicate the front face ofthe polygon, given a fixed orientation (i.e., clockwise oranticlockwise). A convex polygon is a polygon with no interior anglegreater than 180°. A strictly convex polygon is a polygon with nointerior angle greater than or equal to 180°. The origin and directionvectors defining the ray, and the positions of the vertices defining theconvex polygon can be represented with components in a space-coordinatesystem. The space-coordinate system may represent a world space of thescene being rendered, or it may represent an instance space of an objectinstance that is placed within the scene being rendered.

To determine whether a ray intersects a convex polygon, an intersectionpoint of a ray and the plane containing the convex polygon can bedetermined, and then it can be determined whether the intersection pointis inside the polygon. One general approach is to map the vertices(i.e., the coordinates in world or instance space) onto a plane via aprojection parallel to the ray direction. Provided it does not containthe ray, the plane may be freely specified such that both orthographicand oblique projections are permissible. This reduces the problem to a2D problem, and there are many ways of determining whether a 2D point iswithin a 2D convex polygon. For example, where the convex polygon is atriangle, the intersection point can be transformed into a UV coordinatefor the triangle, which makes the intersection test trivial. Problemscan occur if it is determined that the ray intersects a point on theboundary, i.e. an edge or vertex, of the polygon. In particular, oftenobjects are represented with multiple polygons, e.g. with meshes ofpolygons, resulting in shared vertices which define two or more of theconvex polygons. Furthermore, polygons can have shared edges, i.e.,edges where both endpoints are shared vertices. An important type ofshared vertex is one at the centre of a closed fan: A closed fan isdefined as two or more polygons sharing a vertex that are connected byshared edges, such that these shared edges are the only edges with theshared vertex as one of its endpoints. If a ray intersects a point on ashared edge or shared vertex of a closed fan, then, in most cases, it isdesirable for the ray to be found to intersect one or more of thepolygons. If the intersection tests ensure that a ray that intersects apoint on a shared edge or shared vertex of a closed fan intersects atleast one of the polygons then the intersection tests are described asbeing “watertight”. If the intersection tests ensure that a ray thatintersects a point on a shared edge or a shared vertex of a closed fanintersects one (and only one) of the polygons then the intersectiontests are described as being “non-redundantly watertight”, noting thatin this case only a subset of its boundary may be considered part of apolygon. If a ray which intersected a point on a shared edge was foundto intersect zero polygons then it may appear as though the polygon meshhas a hole in it, such that a colour behind the polygon mesh can be seenthrough the mesh in the rendered image (this can occur fornon-watertight intersection tests, but not for watertight intersectiontests). These sorts of rendering errors can be very noticeable, e.g. ifthe colour behind the polygon mesh is significantly different to thecolour of the polygon mesh. Furthermore, if a ray which intersected apoint on a shared edge was found to intersect more than one polygon thenthe colour that is rendered at positions on that shared edge may dependupon the order in which the polygons are tested for intersection, suchthat the rendering may become non-deterministic. These sorts ofrendering errors can be detrimental to the perceived quality of therendered image. Further reasons to have non-redundant watertightintersection tests include: (i) to avoid redundant work, and (ii) toavoid shading discontinuities at any double/multiple hits in transparentobjects resulting from duplicated intersections. The testing of a rayfor intersection with a first polygon is normally performedindependently of the testing of the ray for intersection with a secondpolygon, and it is noted that ensuring watertightness, and specificallynon-redundant watertightness, for the intersection tests is not trivial.

Since intersection tests of rays against convex polygons (e.g.triangles), are performed many times in a ray tracing system, it can bebeneficial to implement the functionality for performing theseintersection tests in dedicated hardware modules, e.g. using fixedfunction circuitry, rather than implementing these intersection testsusing software modules executed on general purpose processing units.Software implementations generally provide more flexibility becausesoftware is more easily altered after it is designed and/or created thanhardware implementations are. However, hardware implementationsgenerally provide more efficient implementations in terms of latency andpower consumption, so if the desired functionality is known in advance,hardware implementations may be preferred over software implementations.When designing a hardware implementation of an intersection testingmodule which is configured for performing intersection testing there aregenerally competing aims of having: (i) a smaller size (i.e. smallersilicon area), (ii) a lower latency, and (iii) lower power consumption.

SUMMARY

This Summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter.

There is provided a method of performing intersection testing of a raywith a convex polygon in a ray tracing system, wherein the ray and theconvex polygon are defined in a 3D space using a space-coordinatesystem, and wherein the ray is defined with a ray origin and a raydirection, the method comprising:

using a ray-coordinate system to perform intersection testing, whereinthe ray-coordinate system has an origin at the ray origin, and whereinthe ray-coordinate system has three basis vectors,

wherein a first of the basis vectors is aligned with the ray direction;and

wherein a second and a third of the basis vectors: (i) are bothorthogonal to the first basis vector, (ii) are not parallel with eachother, and (iii) have a zero as one component when expressed in thespace-coordinate system; and

outputting a result of performing the intersection testing for use bythe ray tracing system.

The method may further comprise translating vertices defining the convexpolygon, for use in performing the intersection testing, by subtractingthe ray origin from the positions of the vertices defining the convexpolygon.

The second and the third of the basis vectors of the ray-coordinatesystem may have a value of ±1 as one component when expressed in thespace-coordinate system.

The first basis vector, S, when expressed with components of thespace-coordinate system may be S=A(D_(x),D_(y),D_(z)), wherein thesecond basis vector, P, when expressed with components of thespace-coordinate system may be P=B(D_(z),0,−D_(x)); and wherein thethird basis vector, Q, when expressed with components of thespace-coordinate system may be Q=C(0,D_(z),−D_(y)); wherein D_(x), D_(y)and D_(z) are components of the ray direction in the space-coordinatesystem and A, B and C are scalar values. B may be a simplified fractionwith D_(x) in its denominator, and C may be a simplified fraction withD_(y) in its denominator.

The first basis vector, S, when expressed with components of thespace-coordinate system may be S=A(D_(x),D_(y),D_(z)), wherein thesecond basis vector, P, when expressed with components of thespace-coordinate system may be

${P = \left( {{\pm 1},0,{\mp \frac{D_{x}}{D_{z}}}} \right)};$and wherein the third basis vector, Q, when expressed with components ofthe space-coordinate system may be

${Q = \left( {0,{\pm 1},{\mp \frac{D_{y}}{D_{z}}}} \right)};$wherein D_(x), D_(y) and D_(z) are components of the ray direction inthe space-coordinate system and A is a scalar value.

The first basis vector, S, when expressed with components of thespace-coordinate system may be S=A(D_(x),D_(y),D_(z)), wherein thesecond basis vector, P, when expressed with components of thespace-coordinate system may be

${P = \left( {{\pm \frac{D_{z}}{D_{x}}},0,{\mp 1}} \right)};$and wherein the third basis vector, Q, when expressed with components ofthe space-coordinate system may be

${Q = \left( {0,{\pm \frac{D_{z}}{D_{y}}},{\mp 1}} \right)};$wherein D_(x), D_(y) and D_(z) are components of the ray direction inthe space-coordinate system and A is a scalar value.

The first basis vector, S, when expressed with components of thespace-coordinate system may be S=A(D_(x),D_(y),D_(z)), wherein thesecond basis vector, P, when expressed with components of thespace-coordinate system may be

${P = \left( {{\pm \frac{1}{D_{x}}},0,{\mp \frac{1}{D_{z}}}} \right)};$and wherein the third basis vector, Q, when expressed with components ofthe space-coordinate system may be

${Q = \left( {0,{\pm \frac{1}{D_{y}}},{\mp \frac{1}{D_{z}}}} \right)};$wherein D_(x), D_(y) and D_(z) are components of the ray direction inthe space-coordinate system and A is a scalar value.

The first basis vector, S, when expressed with components of thespace-coordinate system may be S=A(D_(x),D_(y),D_(z)), wherein thesecond basis vector, P, when expressed with components of thespace-coordinate system may be P=(±D_(z),0,∓D_(x)); and wherein thethird basis vector, Q, when expressed with components of thespace-coordinate system may be Q=(0,±D_(z),∓D_(y)); wherein D_(x), D_(y)and D_(z) are components of the ray direction in the space-coordinatesystem and A is a scalar value.

In some examples

$A = {\frac{1}{❘D_{z}❘}.}$

The method may further comprise transforming the ray and the convexpolygon from the space-coordinate system into the ray-coordinate system,wherein the intersection testing may be performed in the ray-coordinatesystem.

The intersection testing may be performed using fixed functioncircuitry. The fixed function circuitry may comprise one or moremultiply-and-add components for performing multiplications and additionsusing the second and third basis vectors of the ray-coordinate system.

The method may further comprise selectively permuting the components ofthe ray and the vertices defining the convex polygon, such that|D_(z)|≥|D_(x)| and |D_(z)|≥|D_(y)|, before performing intersectiontesting of the ray with the convex polygon.

The method may further comprise selectively reversing the components ofthe ray and the vertices defining the convex polygon, such that D_(x)≥0,D_(y)≥0 and D_(z)≥0, before performing intersection testing of the raywith the convex polygon.

The method may further comprise storing a mapping indication of whetherthe mapping from the space-coordinate system to the ray-coordinatesystem affects the perceived polygon orientation, wherein the mappingindication may be used to perform said intersection testing of the raywith the convex polygon.

Values of at least one of:

$\begin{matrix}{{\frac{D_{x}}{D_{z}}{and}\frac{D_{y}}{D_{z}}},} & (i)\end{matrix}$ $\begin{matrix}{{\frac{D_{z}}{D_{x}}{and}\frac{D_{z}}{D_{x}}},} & ({ii})\end{matrix}$ $\begin{matrix}{{\frac{1}{D_{x}}{and}\frac{1}{D_{y}}},} & ({iii})\end{matrix}$ and $\begin{matrix}\begin{matrix}\frac{1}{D_{z}}\end{matrix} & ({iv})\end{matrix}$or D_(z), may be pre-computed for the ray and stored in a store, whereinthe method may comprise reading the stored values from the store for usein performing intersection testing.

The outputted result may be used in the ray tracing system for renderingan image of a 3D scene.

The convex polygon may be a triangle.

The is provided a method of performing intersection testing of a raywith a convex polygon in a ray tracing system, wherein the ray and theconvex polygon are defined in an n-dimensional space using aspace-coordinate system, wherein n≥3, and wherein the ray is definedwith a ray origin and a ray direction, the method comprising:

using a ray-coordinate system to perform intersection testing, whereinthe ray-coordinate system has an origin at the ray origin, and whereinthe ray-coordinate system has n basis vectors,

-   -   wherein a first of the basis vectors is aligned with the ray        direction,    -   wherein (n−1) of the basis vectors are orthogonal to the first        basis vector,    -   wherein no pairing of the (n−1) basis vectors are parallel with        each other, and    -   wherein the (n−1) basis vectors have zeros for (n−2) components        when expressed in the space-coordinate system; and

outputting a result of performing the intersection testing for use bythe ray tracing system.

There is provided an intersection testing module, for use in a raytracing system, configured to perform intersection testing of a ray witha convex polygon, wherein the ray and the convex polygon are defined ina 3D space using a space-coordinate system, and wherein the ray isdefined with a ray origin and a ray direction, the intersection testingmodule being configured to:

use a ray-coordinate system to perform intersection testing, wherein theray-coordinate system has an origin at the ray origin, and wherein theray-coordinate system has three basis vectors,

wherein a first of the basis vectors is aligned with the ray direction;and

wherein a second and a third of the basis vectors: (i) are bothorthogonal to the first basis vector, (ii) are not parallel with eachother, and (iii) have a zero as one component when expressed in thespace-coordinate system; and

output a result of performing the intersection testing for use by theray tracing system.

The first basis vector, S, when expressed with components of thespace-coordinate system may be S=A(D_(x),D_(y),D_(z)), wherein thesecond basis vector, P, when expressed with components of thespace-coordinate system may be P=B(D_(z),0,−D_(x)); and wherein thethird basis vector, Q, when expressed with components of thespace-coordinate system may be Q=C(0,D_(z),−D_(y)); wherein D_(x), D_(y)and D_(z) are components of the ray direction in the space-coordinatesystem, A is a scalar value, and either:

$\begin{matrix}{{B = {{{\pm \frac{1}{D_{x}}}{and}C} = {\pm \frac{1}{D_{y}}}}},} & (i)\end{matrix}$ or $\begin{matrix}{B = {{{\pm \frac{1}{D_{z}D_{x}}}{and}{}C} = {\pm {\frac{1}{D_{z}D_{y}}.}}}} & ({ii})\end{matrix}$

The intersection testing module may comprise fixed function circuitryconfigured to perform intersection testing using the ray-coordinatesystem. The fixed function circuitry may comprise one or moremultiply-and-add components for performing multiplications and additionsusing the second and third basis vectors of the ray-coordinate system.

The intersection testing module may further comprise a store configuredto store values of at least one of:

$\begin{matrix}{{\frac{D_{x}}{D_{z}}{and}{}\frac{D_{y}}{D_{z}}},} & (i)\end{matrix}$ $\begin{matrix}{{\frac{D_{z}}{D_{x}}{and}\frac{D_{z}}{D_{y}}},} & ({ii})\end{matrix}$ $\begin{matrix}{{\frac{1}{D_{x}}{and}\frac{1}{D_{y}}},} & ({iii})\end{matrix}$ and $\begin{matrix}\frac{1}{D_{z}} & ({iii})\end{matrix}$or D_(z), for the ray, wherein the intersection testing module may beconfigured to read the stored values from the store for use inperforming intersection testing.

There is provided an intersection testing module, for use in a raytracing system, configured to perform intersection testing of a ray witha convex polygon, wherein the ray and the convex polygon are defined inan n-dimensional space using a space-coordinate system, wherein n≥, andwherein the ray is defined with a ray origin and a ray direction, theintersection testing module being configured to:

use a ray-coordinate system to perform intersection testing, wherein theray-coordinate system has an origin at the ray origin, and wherein theray-coordinate system has n basis vectors,

-   -   wherein a first of the basis vectors is aligned with the ray        direction,    -   wherein (n−1) of the basis vectors are orthogonal to the first        basis vector,    -   wherein no pairing of the (n−1) basis vectors are parallel with        each other, and    -   wherein the (n−1) basis vectors have zeros for (n−2) components        when expressed in the space-coordinate system; and

output a result of performing the intersection testing for use by theray tracing system.

There may be provided an intersection testing module configured toperform any of the methods described herein.

There may be provided computer readable code configured to cause any ofthe methods described herein to be performed when the code is run.

There may be provided a method of performing intersection testing, in aray tracing system, for a ray with respect to a plurality of convexpolygons, wherein each of the convex polygons is defined by an orderedset of vertices, and wherein at least one of the vertices is a sharedvertex which is used to define two or more of the convex polygons, themethod comprising:

projecting the vertices of the convex polygons onto a pair of axesorthogonal to the ray direction, wherein the origin of the pair of axescorresponds with the ray origin, and wherein a vertex ordering schemedefines an ordering of the projected vertices which is independent ofthe ordering of the vertices in the ordered sets of vertices definingthe convex polygons; and

for each of the convex polygons:

-   -   for each edge of the convex polygon defined by two of the        projected vertices, determining a parameter indicative of which        side of the edge the ray passes on, wherein if the ray is        determined to intersect a point on the edge then the parameter        is determined based upon whether the ordering, defined by the        vertex ordering scheme, of the projected vertices defining the        edge matches the ordering of the vertices in the ordered set of        vertices defining the convex polygon; and    -   determining whether the ray intersects the convex polygon based        on the parameters determined for the edges of the convex        polygon;

wherein for two or more of the convex polygons the ray is determined tointersect a point on an edge of that convex polygon.

There may be provided an intersection testing module, for use in a raytracing system, configured to perform intersection testing for a raywith respect to a plurality of convex polygons, wherein each of theconvex polygons is defined by an ordered set of vertices, and wherein atleast one of the vertices is a shared vertex which is used to define twoor more of the convex polygons, the intersection testing module beingconfigured to:

project the vertices of the convex polygons onto a pair of axesorthogonal to the ray direction, wherein the origin of the pair of axescorresponds with the ray origin, and wherein a vertex ordering schemedefines an ordering of the projected vertices which is independent ofthe ordering of the vertices in the ordered sets of vertices definingthe convex polygons; and

for each of the convex polygons:

-   -   for each edge of the convex polygon defined by two of the        projected vertices, determine a parameter indicative of which        side of the edge the ray passes on, wherein the intersection        testing module is configured to determine the parameter, if the        ray is determined to intersect a point on the edge, based upon        whether the ordering, defined by the vertex ordering scheme, of        the projected vertices defining the edge matches the ordering of        the vertices in the ordered set of vertices defining the convex        polygon; and    -   determine whether the ray intersects the convex polygon based on        the parameters determined for the edges of the convex polygon.

There may be provided a method of performing intersection testing, in aray tracing system, for a ray with respect to a plurality of convexpolygons, wherein each of the convex polygons is defined by an orderedset of vertices, and wherein at least one of the vertices is a sharedvertex which is used to define two or more of the convex polygons, themethod comprising:

projecting the vertices of the convex polygons onto a pair of axesorthogonal to the ray direction, wherein the origin of the pair of axescorresponds with the ray origin; and

for each of the convex polygons:

-   -   for each edge of the convex polygon defined by two of the        projected vertices, determining a signed parameter, wherein the        sign of the signed parameter is indicative of which side of the        edge the ray passes on, wherein the position of a first of the        projected vertices v_(i) defining the edge is defined with        coordinates p_(i) and q_(i) along the respective axes of the        pair of axes, and wherein a position of a second of the        projected vertices v_(j) defining the edge is defined with        coordinates p_(j) and q_(j) along the respective axes of the        pair of axes, wherein if the ray is determined to intersect a        point on the edge then the sign of the signed parameter is        determined using a module which is configured to:        -   take as inputs, indications which classify each of the            p_(i), q_(i), p_(j) and q_(j) coordinates as negative, zero            or positive, and        -   output, for valid combinations of classifications of the            p_(i), q_(i), p_(j) and q_(j) coordinates, an indication of            the sign of the signed parameter; and    -   determining whether the ray intersects the convex polygon based        on the signs of the signed parameters determined for the edges        of the convex polygon;

wherein for two or more of the convex polygons the ray is determined tointersect a point on an edge of that convex polygon.

There may be provided an intersection testing module, for use in a raytracing system, configured to perform intersection testing for a raywith respect to a plurality of convex polygons, wherein each of theconvex polygons is defined by an ordered set of vertices, and wherein atleast one of the vertices is a shared vertex which is used to define twoor more of the convex polygons, the intersection testing module beingconfigured to:

project the vertices of the convex polygons onto a pair of axesorthogonal to the ray direction, wherein the origin of the pair of axescorresponds with the ray origin; and

for each of the convex polygons:

-   -   for each edge of the convex polygon defined by two of the        projected vertices, determine a signed parameter, wherein the        sign of the signed parameter is indicative of which side of the        edge the ray passes on, wherein the position of a first of the        projected vertices v_(i) defining the edge is defined with        coordinates p_(i) and q_(i) along the respective axes of the        pair of axes, and wherein a position of a second of the        projected vertices v_(j) defining the edge is defined with        coordinates p_(j) and q_(j) along the respective axes of the        pair of axes, wherein the intersection testing module is        configured to use a module to determine the sign of the signed        parameter if the ray is determined to intersect a point on the        edge, wherein the module is configured to:        -   take as inputs, indications which classify each of the            p_(i), q_(i), p_(j) and q_(j) coordinates as negative, zero            or positive, and        -   output, for valid combinations of classifications of the            p_(i), q_(i), p_(j) and q_(j) coordinates, an indication of            the sign of the signed parameter; and    -   determine whether the ray intersects the convex polygon based on        the signs of the signed parameters determined for the edges of        the convex polygon.

The intersection testing module may be embodied in hardware on anintegrated circuit. There may be provided a method of manufacturing, atan integrated circuit manufacturing system, an intersection testingmodule. There may be provided an integrated circuit definition datasetthat, when processed in an integrated circuit manufacturing system,configures the system to manufacture an intersection testing module.There may be provided a non-transitory computer readable storage mediumhaving stored thereon a computer readable description of an intersectiontesting module that, when processed in an integrated circuitmanufacturing system, causes the integrated circuit manufacturing systemto manufacture an integrated circuit embodying an intersection testingmodule.

There may be provided an integrated circuit manufacturing systemcomprising: a non-transitory computer readable storage medium havingstored thereon a computer readable description of the intersectiontesting module; a layout processing system configured to process thecomputer readable description so as to generate a circuit layoutdescription of an integrated circuit embodying the intersection testingmodule; and an integrated circuit generation system configured tomanufacture the intersection testing module according to the circuitlayout description.

There may be provided computer program code for performing any of themethods described herein. There may be provided non-transitory computerreadable storage medium having stored thereon computer readableinstructions that, when executed at a computer system, cause thecomputer system to perform any of the methods described herein.

The above features may be combined as appropriate, as would be apparentto a skilled person, and may be combined with any of the aspects of theexamples described herein.

BRIEF DESCRIPTION OF THE DRAWINGS

Examples will now be described in detail with reference to theaccompanying drawings in which:

FIG. 1 shows a ray tracing system according to examples describedherein;

FIG. 2 shows a ray and two polygons, the basis vectors of aspace-coordinate system, and the basis vectors of a ray-coordinatesystem;

FIG. 3 is a flow chart for a method of performing intersection testingfor a ray with respect to a plurality of convex polygons according toexamples described herein;

FIG. 4 is a flow chart for a first example method of determining aparameter to indicate which side of an edge of a polygon a ray passeson;

FIG. 5 is a flow chart for a second example method of determining aparameter to indicate which side of an edge of a polygon a ray passeson;

FIG. 6 shows a ray intersecting a polygon;

FIG. 7 a shows a ray intersecting a point on an edge that is sharedbetween two polygons, with indications of winding orders of the twopolygons;

FIG. 7 b shows the same situation as that in FIG. 7 a in which a rayintersects a point on an edge that is shared between two polygons, andFIG. 7 b indicates directions of the edges of the two polygons accordingto a vertex ordering scheme;

FIG. 8 a shows a ray intersecting a vertex that is shared between fivepolygons forming a closed fan, with indications of winding orders of thefive polygons;

FIG. 8 b shows the same situation as that in FIG. 8 a in which a rayintersects a vertex that is shared between five polygons forming aclosed fan, and FIG. 8 b indicates directions of the edges of the fivepolygons according to a vertex ordering scheme;

FIG. 9 is a flow chart for a third example method of determining aparameter to indicate which side of an edge of a polygon a ray passeson;

FIG. 10 illustrates a perturbation ε which can be applied to positionsof vertices;

FIGS. 11 a to 11 l illustrate different situations in which a 2D crossproduct can be exactly zero;

FIGS. 12 a to 12 j illustrate further situations in which a 2D crossproduct can be determined to be zero due to underflow when a roundingmode which is not a round away from zero rounding mode is used todetermine the 2D cross product;

FIG. 13 a illustrates a rotation and a reflection of a pair of polygonsin a “winding/orientation dependent” scheme;

FIG. 13 b illustrates a rotation and a reflection of a pair of polygonsin a “winding/orientation independent” scheme;

FIG. 13 c shows a mesh of polygons forming a circular band withindications of which edges are considered to be part of which polygonsin a “winding/orientation dependent” scheme;

FIG. 13 d shows a mesh of polygons forming a circular band withindications of which edges are considered to be part of which polygonsin a “winding/orientation independent” scheme;

FIG. 14 shows a computer system in which a ray tracing system isimplemented; and

FIG. 15 shows an integrated circuit manufacturing system for generatingan integrated circuit embodying a ray tracing system.

The accompanying drawings illustrate various examples. The skilledperson will appreciate that the illustrated element boundaries (e.g.,boxes, groups of boxes, or other shapes) in the drawings represent oneexample of the boundaries. It may be that in some examples, one elementmay be designed as multiple elements or that multiple elements may bedesigned as one element. Common reference numerals are used throughoutthe figures, where appropriate, to indicate similar features.

DETAILED DESCRIPTION

The following description is presented by way of example to enable aperson skilled in the art to make and use the invention. The presentinvention is not limited to the embodiments described herein and variousmodifications to the disclosed embodiments will be apparent to thoseskilled in the art.

Embodiments will now be described by way of example only.

Even when an acceleration structure is used, the amount of work involvedin performing intersection testing in a ray tracing system is still verylarge. For example, ray tracing may be used for rendering an image of a3D scene, where the image may have of the order of a million pixels. Thepixel colour values are derived from some distribution of samples,associated with points in the image plane (typically, there is aone-to-one correspondence between pixel and sample location, but regionsof an image may have a higher or lower sample density or may otherwisebe independent of the arrangement of pixels). In the context of raytracing, the samples are themselves associated with a distribution (inthe statistical sense) of primary rays parameterised by theneighbourhood of each sample location. In the simplest example, a singleprimary ray is traced for each sample and used to determine its result.In other examples, multiple rays may be generated in accordance with thedistribution (e.g. stochastic sampling) and the result derived from someaccumulation or combination of the individual primary rays. When it isdetermined that a ray intersects with an object in the scene, a shadercan be executed which may result in the emission of another ray (i.e. a“secondary ray”) into the scene. Each primary ray may result in theemission of many secondary rays, which are all traced through the sceneto determine their intersections. Therefore, it would not be unusual forthere to be tens or hundreds of millions of rays traced through a scenefor rendering an image. The complexity of scenes to be rendered tends toincrease as graphics rendering technology develops, so it would not beunusual for there to be thousands of objects in a scene, each of whichmay be represented by many primitives (e.g. polygons, such astriangles). Furthermore, the images being rendered may represent framesof a sequence of frames which are to be rendered in real-time, e.g. fora display to a user in real-time. For example, the user may be playing agame wherein the rendered images represent a users view of the 3D sceneas the user plays the game. In order for the sequence of frames toappear like a continuous stream of video data, many frames may berendered per second, e.g. 24, 30 or 60 frames per second to give someexamples. It can therefore be appreciated that the work involved inperforming intersection testing in a ray tracing system to render scenesto be output in real-time is vast.

One way to overcome this problem, and to perform ray tracing to renderscenes to be output in real-time would be to have one or moresupercomputers to perform all of the processing. This could beconsidered to be a ‘brute force’ approach. However, as well as an aim tohave high performance (to perform ray tracing to render scenes to beoutput in real-time), there are also competing aims of reducing the size(e.g. silicon area) and power consumption of the ray tracing system. Forexample, there may be an aim to implement the ray tracing system on amobile device, such as a tablet or smartphone, for which the acceptablesize and power consumption may be much lower than for a supercomputer.As such, when designing a ray tracing system, there may be a trade-offbetween performance, power consumption and area. Depending on how thistrade-off is implemented, examples described herein may allow theperformance to be increased without a significant increase to the powerconsumption and area (compared to the prior art described above in thebackground section). Alternatively, in a different implementation of thetrade-off, examples described herein may allow the power consumptionand/or size of the ray tracing system to be decreased withoutsignificantly decreasing the performance of the ray tracing system(compared to the prior art described above in the background section).Different implementations can be designed to target different points inthe trade-off between performance, power consumption and silicon area.

As described above, since objects are often represented as sets ofconvex polygons (non-convex polygons can be described in terms of unionsof convex polygons), testing rays for intersection with convex polygons(e.g. triangles) is an extremely frequent operation in a ray tracingsystem. Therefore, any optimizations that can be made to the way inwhich the intersection tests are performed can be very useful foroptimizing the ray tracing system in terms of reducing the latency,power consumption and size of the ray tracing system.

Furthermore, since such objects typically describe connected surfaceswhereby neighbouring polygons have one or more shared vertices or edges,it is important that gaps do not spuriously appear as ray intersectionssweep over the object interior, producing noticeable structural defects(which may be referred to as “rendering artefacts” in the renderedimage). These may manifest due to rounding error or otherwiseapproximation in the intersection testing of each polygon wherebyimplicit perturbation of common vertices and/or edges may renderneighbouring polygons disconnected when expressed in terms of theirindividual intersection results. An implementation that prevents anysuch disconnection from occurring may be referred to as “watertight”.

While it is possible to mitigate the appearance of gaps by fattening theboundary of each convex polygon to some extent in order to provide someerror tolerance, it is also desirable for an implementation to exhibitsome form of non-redundant watertight behaviour i.e. for it to minimisethe number of reported intersections that occur between a single ray anda single object. It is therefore preferable for an implementation toprovide consistent results across individual polygon intersection tests,especially those with common vertices and/or edges, without modificationof the polygon boundaries. However, it is usually not feasible toimplement fully accurate tests in the general case so we are left withhaving to ensure that whatever errors are introduced are compatible withthe aim of ensuring non-redundant watertightness. If we limit ourselvesto cases in which an edge is only considered common if it is bounded bya pair of common vertices (so that “T-junctions” are not present) thenit is possible to achieve this by decomposing each intersection testinto a projection of the polygon vertices onto a plane (not containingthe ray), followed by an accurate intersection test in the plane. Thisseries of operations consolidates all of the error into per-vertexcontributions, ensuring that results are consistent across connectedtriangles since common vertices remain common after transformation(thereby preserving the local topology) and the subsequent intersectiontests are exact.

In the context of the above operations, an implementation may then bereferred to as “non-redundantly watertight” if and only if a singlepositive intersection result is reported for any set of (exact) commonvertex or edge intersections in the plane. To put this another way,intersection testing of rays with convex polygons is “watertight” if itis ensured that a ray that intersects a point on a shared edge ofmultiple polygons or a shared vertex of a closed fan is determined tointersect at least one of the polygons which share the edge or vertex.Furthermore, intersection testing of rays with convex polygons is“non-redundantly watertight” if it is ensured that a ray that intersectsa point on a shared edge of multiple convex polygons or a shared vertexof a closed fan is determined to intersect a single one of the polygonswhich share the edge or vertex (i.e. the ray is determined to intersectone and only one of the polygons which share the edge or vertex).

Examples described herein further qualify this watertightness propertyby stipulating that common edges and vertices belong to polygons withconsistent orientation, either with respect to each other(distinguishing outward and inward facing polygons i.e. in accordancewith their winding order) or with respect to the perspective of the ray(distinguishing front facing and back facing polygons). In fact, thereare several (non-mutually exclusive) scenarios (or any combinationthereof) where the properties of “watertight” and/or “non-redundantlywatertight” may or may not apply: (i) shared vertices not of a closedfan (e.g., boundary vertices of a mesh); (ii) non-manifold geometry,i.e., edges shared by three or more polygons; (iii) shared edges orshared vertices of a closed fan with polygons specified withinconsistent winding orders (i.e., the endpoints of any shared edge aregiven in the same order); (iv) shared edges or shared vertices of aclosed fan with polygons, specified with consistent winding orders (i.e.the endpoints of any shared edge are given in the opposite order), withmore than one orientation (i.e., clockwise and anticlockwise) from theviewpoint of the ray (e.g., silhouette edges or vertices); (v)non-simple geometry, i.e., intersections between polygons not at sharededges or shared vertices (e.g. interior intersections, T-junctions); and(vi) redundant geometry, i.e., intersections of shared edges or sharedvertices of a closed fan shared by degenerate polygons. In case (vi), itis desirable for the properties of “watertight” or “non-redundantlywatertight” to hold, but only when the degenerate polygons are omitted.

In the examples described herein the vertices of the convex polygons areprojected onto a pair of axes orthogonal to the ray direction, whereinthe origin of the pair of axes corresponds with the ray origin. In thisway, the intersection testing process for testing whether a rayintersects a convex polygon in a 3D scene is reduced to a 2D problemwhich, as described above, enables us to ensure watertight intersection.The pair of axes are not parallel with each other. The ray is located atthe origin of the pair of axes. The direction vector of the ray has amagnitude of zero along both of the axes in the pair of axes. Therefore,the ray is determined to intersect a polygon only if the polygon coversthe origin of the pair of axes, where the ray may or may not intersect apolygon at its boundary. Different examples for ensuring watertightnessof the intersection testing when the ray intersects a point on an edgeof a polygon are described herein.

In examples described herein the pair of axes represent two basisvectors of a ray coordinate system. The particular ray coordinatesystems described herein are particularly beneficial for use inperforming intersection testing in a ray tracing system, although inother examples the pair of axes might not be part of the particular raycoordinate systems described herein.

For example, in the ray coordinate systems described herein, some of thecomponents of the basis vectors, when expressed in the space-coordinatesystem, are zero. Furthermore, the ray may be rescaled such that thelargest component(s) of the ray direction vector have unit magnitude.The choice of the ray coordinate system and the rescaling of the ray inexamples described herein may make two thirds of the testsmathematically cheaper due to scale values being either 1.0 or 0.0. Thiscan be achieved by performing a small amount of pre-calculation that isconstant for the ray, so the cost of performing this pre-calculation canbe ameliorated because it is performed once for a ray and then can beused in many intersection tests involving that ray. For example, wherethe ray direction vector D has components D_(x), D_(y) and D_(z), valuesof

$\frac{D_{x}}{D_{z}}{and}\frac{D_{y}}{D_{z}}$(or values of

$\left. {\frac{D_{x}}{❘D_{z}❘}{and}\frac{D_{y}}{❘D_{z}❘}} \right)$for a ray may be pre-computed and may be stored. As described in moredetail below, the axes can be swapped (e.g. permuted) to ensure thatD_(z) is the major component of the ray direction vector, i.e.|D_(z)|≥|D_(z)| and |D_(z)|≥|D_(y)|. Since any valid ray directionvector has a non-zero magnitude this means that |D_(z)|>0. As anotherexample, values of

$\frac{D_{z}}{D_{x}}{and}\frac{D_{z}}{D_{y}}$(or values of

$\left. {\frac{❘D_{z}❘}{D_{x}}{and}\frac{❘D_{z}❘}{D_{y}}} \right)$for a ray may be pre-computed and may be stored, with D_(x) and D_(y)nonzero, or by using the concept of signed infinity, e.g. as defined inthe IEEE floating point format, if D_(x) or D_(y) is zero. In someexamples, values of zero can be replaced with non-zero values that aresmall enough to behave like zero in the operations described herein,and/or values of infinity can be replaced with finite values that arelarge enough to behave like infinity in the operations described herein,and these examples avoid having to treat some zeros and infinities asspecial cases, and also avoid at least some undefined results which canresult from multiplying or dividing by zero or infinity. It is notedthat the values of

$\frac{D_{z}}{D_{x}}{and}\frac{D_{z}}{D_{y}}$(or values of

$\left. {\frac{❘D_{z}❘}{D_{x}}{and}\frac{❘D_{z}❘}{D_{y}}} \right)$have a modulus greater than or equal to 1. As another example, a valueof

$\frac{1}{D_{z}}$for a ray may be pre-computed and may be stored. After these values havebeen pre-computed they can be used for performing intersection testingon the ray. The pre-computed values can be used for multipleintersection tests, which may be performed in parallel. In someexamples, the pre-computed values are stored so that they can be read,rather than calculated for use in performing intersection testing on theray. Furthermore, rescaling the ray so that the largest component(s) ofthe ray direction vector have unit magnitude and/or the choice of theray coordinate system so that one or more of the components of the basisvectors are zero makes the processing of the rays simpler to implement.Reducing the number of tests that need to be performed in order todetermine whether a ray intersects a convex polygon, and/or simplifyingthe processing of the rays for determining whether a ray intersects aconvex polygon, can reduce the latency, size and/or power consumption ofan intersection testing module in a ray tracing system.

FIG. 1 shows a ray tracing system 100 comprising a ray tracing unit 102and a memory 104. The ray tracing unit 102 comprises a processing module106, an intersection testing module 108 and processing logic 110. Theintersection testing module 108 comprises one or more box intersectiontesting units 112, one or more polygon intersection testing units 114, aray rescaling unit 116 and an instance transform unit 118. It is notedthat there may be more than one ray rescaling unit and/or more than oneinstance transform unit. In operation the ray tracing unit 102 receivesgeometric data defining objects within the 3D scene. The ray tracingunit 102 also receives ray data defining rays that are to be tested forintersection. The rays may be primary rays or secondary rays. Theprocessing module 106 is configured to generate an accelerationstructure based on the geometric data, and to send the accelerationstructure to the memory 104 for storage therein. After the accelerationstructure has been stored in the memory 104, the intersection testingmodule 108 can retrieve nodes (e.g. comprising data definingaxis-aligned boxes corresponding to the nodes) of the accelerationstructure from the memory 104 to perform intersection testing of raysagainst the retrieved nodes. To avoid reading in the whole accelerationstructure at a time, the intersection testing module 108 retrieves asubset of the boxes from one level of the acceleration structure frommemory 104 at each stage, based on the results of previous intersectiontests. The box intersection testing unit(s) 112 perform intersectiontests to determine whether or not a ray intersects each of the boundingboxes corresponding to nodes of the acceleration structure (where a misscan cull vast swathes of the hierarchical acceleration structure). If itis determined that a leaf node is intersected then the polygonintersection testing unit(s) 114 perform one or more polygonintersection tests to determine which object(s) (if any) the rayintersects. In this example, the convex polygons are triangles, althoughit is noted that in other examples, the convex polygons could be othershapes such as squares, rectangles, pentagons, hexagons, etc., so longas they are planar. The results of the intersection tests indicate whichobject in the scene a ray intersects, and the results may also indicatea position on the object at which the ray intersects the object (e.g.Barycentric coordinates), may also indicate a distance along the raythat the intersection occurs, and may also indicate the perceivedorientation of the object from the ray's point of view (e.g.,clockwise/anticlockwise or front/back facing). In some instances, theintersection determination may be based on whether the distance alongthe ray that the intersection occurs is between the minimum and maximumclipping distances for the ray (t_(min) and t_(max)). In some instances,the intersection determination may be based on the perceived orientationof the object (e.g., for back-face culling). The results of theintersection testing are provided to the processing logic 110. Theprocessing logic 110 is configured to process the results of theintersection testing to determine rendered values representing the imageof the 3D scene. The rendered values determined by the processing logic110 can be passed back to the memory 104 for storage therein torepresent the image of the 3D scene.

In the examples described herein the ray tracing system uses anacceleration structure in order to reduce the number of intersectiontests that need to be performed for a ray against convex polygons.However, it is noted that some other examples might not use anacceleration structure, and may simply tests rays against the convexpolygons without first attempting to reduce the number of intersectiontests that need to be performed using an acceleration structure.

Generally, when performing intersection testing of a ray with a convexpolygon, the ray is defined in terms of components in a space-coordinatesystem in which the convex polygon is defined. However, in examplesdescribed herein, a ray-coordinate system is derived relative to the rayitself, and the convex polygon can be mapped onto the ray coordinatesystem. The term “space-coordinate system” is used herein to refer tothe coordinate system in which the convex polygon is defined. The“space-coordinate system” may be a world space-coordinate system or aninstance space-coordinate system. In most of the examples describedherein the space-coordinate system is a three-dimensional coordinatesystem, and a convex polygon is a region of three-dimensional spacecontained entirely within a plane, bounded by n≥3 line segments, suchthat all interior angles are less than or equal to 180°. A strictlyconvex polygon is a convex polygon such that all interior angles arestrictly less than 180°.

The term “ray-coordinate system” is used herein to refer to a coordinatesystem that is specific to a ray, and which has its origin at the originof the ray. It is noted that in the examples described in detail hereinthe origin of the ray-coordinate system is the origin of the ray, but inother examples, any point along the ray's line could be used as theorigin of the ray-coordinate system, with a suitable adjustment to theminimum and maximum clipping distances for the ray (t_(min) andt_(max)). The ray-coordinate system has three basis vectors. A first ofthe basis vectors is aligned with the ray direction. A second and thirdof the basis vectors are both orthogonal to the first basis vector, andare not parallel with each other. In examples described herein thesecond and third basis vectors of the ray-coordinate system are not, ingeneral, orthogonal to each other, although in some examples it ispossible that they are orthogonal to each other. Furthermore, inexamples described herein, the second and third basis vectors of theray-coordinate system have a 0 as one component when expressed in thespace-coordinate system. In some examples, the second and third of thebasis vectors of the ray-coordinate system have a value of ±1 (i.e. amagnitude of 1) for one component when expressed in the space-coordinatesystem.

In examples described herein, the ray-coordinate system is used by theintersection testing module 108 to perform intersection testing todetermine whether a ray intersects a convex polygon, wherein the ray andthe convex polygon are defined in a 3D space using a space-coordinatesystem. A result of performing the intersection testing for the ray isoutput from the intersection testing module 108 for use by the raytracing system, e.g. for use in determining which shader program(s)is(are) executed for the ray by the processing logic 110.

The vertices defining the convex polygon are translated for use inperforming the intersection testing using the ray-coordinate system bysubtracting the ray origin from the positions of the vertices definingthe convex polygon. Subtracting the ray origin from the positions of thevertices defining the convex polygon means that the position of thepolygon is then defined relative to the origin of the ray. By shiftingthe vertices by the ray origin first, all relative error in downstreamcalculations is centred around the ray origin. Assuming that the same istrue for the box tester, this makes guaranteeing conservatism (i.e., nofalse negatives) easier.

FIG. 2 illustrates a ray 202 and two convex polygons 204 and 206. Thetwo polygons are triangles and they share an edge such that they form aquad. The orientations of the two polygons are indicated as clockwise,as viewed and from the ray's perspective, in FIG. 2 . Polygon 204 isdefined by the ordered set of vertices (v₀, v₁, v₂), and polygon 206 isdefined by the ordered set of vertices (v₁, v₃, v₂). The edge betweenvertices v₁ and v₂ is shared by polygons 204 and 206. As v₁ and v₂ aredefined in the opposite order in polygons 204 and 206, and therefore theedge has opposite direction in one from the other, the two polygons aresaid to be specified with consistent winding. As described above, theray 202, r(t), can be represented as r(t)=O+Dt, where O is a vectorrepresenting the origin of the ray and D is the direction vector of theray, where O=(O_(x),O_(y),O_(z)) and D=(D_(x),D_(y),D_(z)). The x, y andz basis vectors of the space-coordinate system are shown in FIG. 2 . Asdescribed above, the space-coordinate system could be a world spacecoordinate system of a scene being rendered, or the space-coordinatesystem could be the instance space-coordinate system of an instanced setof geometry within a scene being rendered. The origin of theray-coordinate system that is used in the intersection testing is theorigin of the ray. The positions of the vertices defining the polygons204 and 206 are translated by subtracting the ray origin from thepositions of the vertices. In the example shown in FIG. 2 , the ray 202intersects the polygon 204 at an intersection point 208.

Furthermore, in some examples described herein, the x, y and zcomponents of the ray and the vertices defining the polygons areselectively permuted, such that |D_(z)|≥|D_(x)| and |D_(z)|≥|D_(y)|,before performing intersection testing (noting that in these examples wemust also have |D_(z)|≥0 for valid ray directions). The selectivepermutation of the axes is performed such that D_(z) will be the majorcomponent of the ray direction.

FIG. 2 shows the ray 202 and polygons 204 and 206 in relation to thebasis vectors after the translation and the selective permutation of theaxes of the ray and polygons have been performed, such that the originof the ray is at the origin of the coordinate system, and such that|D_(z)|≥|D_(x)| and |D_(z)|≥|D_(y)|. This ensures that D_(z) is non-zerobecause a valid ray cannot have D_(x)=D_(y)=D_(z)=0.

The basis vectors of the ray-coordinate system are represented as P, Qand S in FIG. 2 . The first basis vector, S, of the ray-coordinatesystem is set to be along the direction of the ray,S=A(D_(x),D_(y),D_(z)), where A is a scalar value. In some examples,

${A = \frac{1}{D_{z}}},$such that

$S = {\left( {\frac{D_{x}}{D_{z}},\frac{D_{y}}{D_{z}},1} \right).}$

As an example (which may be referred to as a “first example” below), thesecond and third basis vectors, P and Q, may be defined to be

$P = {{\left( {1,0,{- \frac{D_{x}}{D_{z}}}} \right){and}{}Q} = {\left( {0,1,{- \frac{D_{y}}{D_{z}}}} \right).}}$more generally, and as shown in the example in FIG. 2 ,P=B(D_(z),0,−D_(x)) and Q=C(0,D_(z),−D_(y)), where B and C are anynon-zero scalar values. To give some other examples, B could be

${\pm 1},{\pm \frac{1}{D_{z}}},{\pm \frac{1}{❘D_{z}❘}},{\pm \frac{1}{D_{x}}},{\pm \frac{1}{❘D_{x}❘}},{\pm {{sgn}\left( D_{z} \right)}},$or ±sgn(D_(x)) or any product of these non-zero scalar values e.g.

$\frac{1}{D_{x}D_{z}}$and C could be

${\pm 1},{\pm \frac{1}{D_{z}}},{\pm \frac{1}{❘D_{z}❘}},{{{\pm \frac{1}{D_{y}}}{or}} \pm \frac{1}{❘D_{y}❘}},{\pm {{sgn}\left( D_{z} \right)}},$or ±sgn(D_(y)) or any product of these non-zero scalar values e.g.

$\frac{1}{D_{y}D_{z}}.$It is noted that, for a non-zero value α, sgn(α)=+1 if α is positive,and sgn(α)=−1 if α is negative. If α=0 then sgn(α) may be +1 or −1depending on the sign bit of the floating point representation of α. Itis noted that standard floating point representations allow both +0 and−0 to be represented separately, and that unsigned 0 is usuallyidentified with +0. In other words, if the adopted number systemdistinguishes between −0 and +0 (as is the case for the IEEE floatingpoint system) then, as an example, sgn(+0)=+1 and sgn(−0)=−1, otherwisesgn(0)=1. A set of common values for A might be the union of the sets ofcommon values for B and C. It is noted that S is orthogonal to P and toQ, which can be seen in that P.S=Q.S=0. However, depending on the valuesof D_(x) and D_(y), P and Q are not necessarily orthogonal to eachother. For example, unless D_(x) or D_(y) is zero then P and Q will notbe orthogonal to each other. As described above, P and Q are notparallel to each other, e.g. P and Q are at least far enough from beingparallel to each other to not cause issues due to loss in rounding,otherwise the system might degenerate into a 1D scenario. It is notedthat D_(z) cannot be zero because of the selective permutation of theaxes such that D_(z) is the major component of the ray direction andbecause a valid ray direction vector must have a non-zero magnitude inorder to define a line. Therefore, of the possible components of raybasis vector P and Q, the values of

$\frac{D_{x}}{D_{z}}{and}\frac{D_{y}}{D_{z}}$are always well-defined and have a magnitude in the range from zero toone inclusive, and the values of

$\frac{D_{z}}{D_{x}}{and}\frac{D_{z}}{D_{y}}$have a magnitude in the range from one to positive infinity (inclusive).

The choice of the scalar values A, B and C can affect the handedness ofthe system. In FIG. 2 , the world (or instance) space axes form aleft-handed system. If we order the ray-space axes as (X′, Y′, Z′)=(P,Q, S), then they form a system of the same handedness (this will be thecase if and only if the major axis is positive). Each scalar value (fromthe scalar values A, B and C) that is negative flips the handedness ofthe system, e.g. from left-handed to right-handed. If an even number ofthe scalar values A, B and C are negative, then they cancel out and wereturn to a left-handed system.

A sign of the mapping from the space-coordinate system to theray-coordinate system may affect the perceived primitive orientation(given a fixed primitive winding). For example, a negative sign of themapping may reverse primitive orientation and a positive sign of themapping may preserve primitive orientation. As an example, the sign ofthe mapping may be given as:sgn(D _(z)) XOR sgn(A) XOR sgn(B) XOR sgn(C) XOR sgn(σ)  (1)wherein sgn(σ) indicates whether the permutation reverses the mappingsign.

The individual terms are XORed together as reversing the systemhandedness twice results in the original handedness. For example, ifA=1/|D_(z)|, B=sgn(D_(z))/D_(x), C=sgn(D_(z))/D_(y), then expression (1)becomes sgn(D_(x)) XOR sgn(D_(y)) XOR sgn(D_(z)) XOR sgn(σ).

The value of sgn(D_(z)) comes from the sign of the determinant of achange-of-basis matrix (i.e. the three ray-coordinate axes as rows orcolumns with A=B=C=1 and the major component already permuted to thefinal row or column), which gives the signed volume of a transformedunit cube, equal to D_(z)(D_(x) ²+D_(y) ²+D_(z) ²)=D_(x)∥D∥². Being themajor component, the magnitude of D_(z) is at least

${\frac{1}{\sqrt{3}}{D}},$hence the magnitude of the determinant is at least

$\frac{1}{\sqrt{3}}{{D}^{3}.}$After normalising this result by the product of the basis vectormagnitudes, this gives a measure of how far the matrix is from singular,i.e. how far the ray-coordinate system is from degenerate, and hencealso how far from parallel the orthogonal axes are. It is beneficial forD to be of such magnitude that these expressions do not underflow oroverflow, to avoid potential precision issues when transforming verticesinto the ray-coordinate system.

In a higher n-dimensional construction, this determinant generalises toD_(z) ^((n−2))∥D∥², both demonstrating that the construction works forall n>1 and that Dz only affects sign when n is odd.

The values of sgn(A), sgn(B) and sgn(C) come from rescaling the aboveray-coordinate system bases. The value of the sign of the permutationsigma, σ, comes from the way the ray components are reordered (i.e.“permuted” by sigma, σ) to put the major component into a specificposition (e.g. the final/z coordinate). By doing so, we leave twochoices for which positions the minor coordinates go in. One of thesechoices corresponds to rotating all the coordinates, the othercorresponds to transposing two of the coordinates. For example, tosimplify muxing, the permutation involves a transposition when Y is themajor component and the permutation involves a rotation otherwise. Forexample, if X is the major component of the ray direction vector thenthe permutation of the axes is the rotation (x,y,z)−>(y,z,x); if Y isthe major component of the ray direction vector then the permutation ofthe axes is the transposition (x,y,z)−>(x,z,y); and if Z is the majorcomponent of the ray direction vector then the permutation of the axesis the identity (x,y,z)−>(x,y,z). This example provides for simplemultiplexing because there are only two inputs for the first twocomponents. Transposing two of the coordinates results in a sign changeto the determinant as it is equivalent to swapping two rows in thechange-of-basis matrix.

In another example (not shown in the Figures, and which is referred toas a “second example” below), the second and third basis vectors, P andQ, are defined to be

$P = {{\left( {{- 1},0,{+ \frac{D_{x}}{D_{z}}}} \right){and}Q} = {\left( {0,{- 1},{+ \frac{D_{y}}{D_{z}}}} \right).}}$In this example (not shown in FIG. 2 ),

${B = {C = {- \frac{1}{D_{z}}}}}.$Generally, these two examples can be described as the second basisvector, P, being defined to be

$P = {\left( {1,0,{- \frac{D_{x}}{D_{z}}}} \right){or}}$${P = \left( {{- 1},0,{+ \frac{D_{x}}{D_{z}}}} \right)},$i.e. either

${P = \left( {{\pm 1},0,{\mp \frac{D_{x}}{D_{z}}}} \right)},$and the third basis vector, Q, being defined to be

${Q = \left( {0,{\pm 1},{\mp \frac{D_{y}}{D_{z}}}} \right)},$i.e. either

$Q = {{\left( {0,1,{- \frac{D_{y}}{D_{z}}}} \right){or}Q} = {\left( {0,{- 1},{+ \frac{D_{y}}{D_{z}}}} \right).}}$

In another example (not shown in the Figures, and which is referred toas a “third example” below),

$B = {{{\pm \frac{1}{❘D_{z}❘}}{and}C} = {\pm {\frac{1}{❘D_{z}❘}.}}}$For example, the second basis vector, P, may be defined to be

${P = \left( {{\pm {{sgn}\left( D_{z} \right)}},0,{{\mp {{sgn}\left( D_{z} \right)}}\frac{D_{x}}{D_{z}}}} \right)},$i.e. either

$P = {\left( {{+ {{sgn}\left( D_{z} \right)}},0,{{- {{sgn}\left( D_{z} \right)}}\frac{D_{x}}{D_{z}}}} \right){or}}$${P = \left( {{- {{sgn}\left( D_{z} \right)}},0,{{+ {{sgn}\left( D_{z} \right)}}\frac{D_{x}}{D_{z}}}} \right)},$and the third basis vector, Q, may be defined to be

${Q = \left( {0,{\pm {{sgn}\left( D_{z} \right)}},{{\mp {{sgn}\left( D_{z} \right)}}\frac{D_{y}}{D_{z}}}} \right)},$i.e. either

$Q = {\left( {0,{+ {{sgn}\left( D_{z} \right)}},{{- {{sgn}\left( D_{z} \right)}}\frac{D_{y}}{D_{z}}}} \right){or}}$$Q = {\left( {0,{- {{sgn}\left( D_{z} \right)}},{{+ {{sgn}\left( D_{z} \right)}}\frac{D_{y}}{D_{z}}}} \right).}$It is noted that

$\begin{matrix}{{{{{sgn}\left( D_{i} \right)}D_{i}} = {❘D_{i}❘}},{{{{sgn}\left( D_{i} \right)}\frac{1}{D_{i}}} = \frac{1}{❘D_{i}❘}},{\frac{D_{i}}{{sgn}\left( D_{i} \right)} = {❘D_{i}❘}},} \\{and} \\{\frac{1}{{{sgn}\left( D_{i} \right)}D_{i}} = {\frac{1}{❘D_{i}❘}.}}\end{matrix}$

In another example (not shown in the Figures, and which is referred toas a “fourth example” below), B=±1 and C=±1. For example, the secondbasis vector, P, may be defined to be P=(±D_(z),0,∓D_(x)), i.e. eitherP=(+D_(z),0,−D_(x)) or P=(−D_(z),0,+D_(x)), and the third basis vector,Q, may be defined to be Q=(0,±D_(z),∓D_(y)), i.e. eitherQ=(0,+D_(z),−D_(y)) or Q=(0,−D_(z),+D_(y)).

In another example (not shown in the Figures, and which is referred toas a “fifth example” below),

$B = {{{\pm \frac{1}{❘D_{x}❘}}{and}C} = {\pm {\frac{1}{❘D_{y}❘}.}}}$For example, the second basis vector, P, may be defined to be

$\begin{matrix}{{P = \left( {{{\pm {{sgn}\left( D_{x} \right)}}\frac{D_{z}}{D_{x}}},0,{{\mp {sgn}}\left( D_{x} \right)}} \right)},} \\{{{i.e.{either}}P} = \left( {{{+ {{sgn}\left( D_{x} \right)}}\frac{D_{z}}{D_{x}}},0,{- {{sgn}\left( D_{x} \right)}}} \right)} \\{{{{or}P} = \left( {{- {{sgn}\left( D_{x} \right)}\frac{D_{z}}{D_{x}}},0,{+ {{sgn}\left( D_{x} \right)}}} \right)},}\end{matrix}$and the third basis vector, Q, may be defined to be

${Q = \left( {0,{{\pm {{sgn}\left( D_{y} \right)}}\frac{D_{z}}{D_{y}}},{\mp {{sgn}\left( D_{y} \right)}}} \right)},$i.e. either

$\begin{matrix}{Q = \left( {0,{{+ {sgn}}\left( D_{y} \right)\frac{D_{z}}{D_{y}}},{- {{sgn}\left( D_{y} \right)}}} \right)} \\{or} \\{Q = {\left( {0,{- {{sgn}\left( D_{y} \right)}\frac{D_{z}}{D_{y}}},{+ {{sgn}\left( D_{y} \right)}}} \right).}}\end{matrix}$

In other examples (not shown in the Figures, and which are referred toas “sixth examples” below),

$B = {{{\pm \frac{1}{D_{x}}}{and}C} = {\pm {\frac{1}{D_{y}}.}}}$Therefore, the second basis vector, P, may be defined to be

${P = \left( {{\pm \frac{D_{z}}{D_{x}}},0,{\mp 1}} \right)},$i.e. either

${P = {{\left( {{+ \frac{D_{z}}{D_{x}}},0,{- 1}} \right){}{or}{}P} = \left( {{- \frac{D_{z}}{D_{x}}},0,{+ 1}} \right)}},$and the third basis vector, Q, may be defined to be

${Q = \left( {0,{\pm \frac{D_{z}}{D_{y}}},{\mp 1}} \right)},$i.e. either

$Q = {{\left( {0,{+ \frac{D_{z}}{D_{y}}},{- 1}} \right){}{or}Q} = {\left( {0,{- \frac{D_{z}}{D_{y}}},{+ 1}} \right).}}$Since D_(z) is the major component of the ray direction vector, it ispossible that D_(x) or D_(y) may be zero. Therefore, in these examples,care needs to be taken when handling the values of

${{\pm \frac{D_{z}}{D_{x}}}{and}} \pm {\frac{D_{z}}{D_{y}}.}$In one approach, me values of D_(x) or D_(y) may be perturbed by somesmall amount so that they are never exactly zero. For example, values ofzero can be replaced with non-zero values that are small enough tobehave like zero in the operations described herein.

In other examples (not shown in the Figures, and which are referred toas “seventh examples” below),

$B = {{{\pm \frac{1}{D_{x}D_{z}}}{and}C} = {\pm {\frac{1}{D_{y}D_{z}}.}}}$Therefore, the second basis vector, P, may be defined to be

${P = \left( {{\pm \frac{1}{D_{x}}},0,{\mp \frac{1}{D_{z}}}} \right)},$i.e. either

${P = {{\left( {{+ \frac{1}{D_{x}}},0,{- \frac{1}{D_{z}}}} \right){}{or}P} = \left( {{- \frac{1}{D_{x}}},0,{+ \frac{1}{D_{z}}}} \right)}},$and the third basis vector, Q, may be defined to be

${Q = \left( {0,{\pm \frac{1}{D_{y}}},{\mp \frac{1}{D_{z}}}} \right)},$i.e. either

$Q = {{\left( {0,{+ \frac{1}{D_{y}}},{- \frac{1}{D_{z}}}} \right){or}Q} = {\left( {0,{- \frac{1}{D_{y}}},{+ \frac{1}{D_{z}}}} \right).}}$Since D_(z) is the major component of the ray direction vector, it ispossible that D_(x) or D_(y) may be zero. Therefore, in these examples,care needs to be taken when handling the values of

${{\pm \frac{1}{D_{x}}}{and}} \pm {\frac{1}{D_{y}}.}$As described above, in one approach, the values of D_(x) or D_(y) may beperturbed by some small amount so that they are never exactly zero. Forexample, values of zero can be replaced with non-zero values that aresmall enough to behave like zero in the operations described herein.

The example basis vectors given in the second, third, fourth and fifthexamples described above all have consistent handedness, whereas thehandedness of the basis vectors given in the sixth and seventh examplesdescribed above may either have consistent or opposite handedness,depending on the signs of D_(x) and D_(y).

In another example (not shown in the Figures), B=±sgn(D_(z)) andC=±sgn(D_(z)). For example, the second basis vector, P, may be definedto be P=(±|D_(z)|,0,±sgn(D_(z))D_(x)), i.e. eitherP=(+|D_(z)|,0,−sgn(D_(z))D_(x)) or P=(−|D_(z)|,0,+sgn(D_(z))D_(x)), andthe third basis vector, Q, may be defined to beQ=(0,±|D_(z)|,±sgn(D_(z))D_(y)), i.e. eitherQ=(0,+|D_(z)|,−sgn(D_(z))D_(y)) or Q=(0,−|D_(z)|,+sgn(D_(z))D_(y)).

In another example (not shown in the Figures), B=±sgn(D_(x)) andC=±sgn(D_(y)). For example, the second basis vector, P, may be definedto be P=(±sgn(D_(x))D_(z),0,∓|D_(x)|), i.e. eitherP=(+sgn(D_(x))D_(z),0,−D_(x)|) or P=(−sgn(D_(x))D_(z),0,+|D_(x)|), andthe third basis vector, Q, may be defined to beQ=(0,±sgn(D_(y))D_(z),∓|D_(y)|), i.e. eitherQ=(0,+sgn(D_(y))D_(z),−|D_(y)|) or Q=(0,−sgn(D_(y))D_(z),+|D_(y)|).

In all of these examples: (i) S is orthogonal to P and to Q, (ii) P andQ are not parallel with each other, and (iii) P and Q have a zero as onecomponent when expressed in the space-coordinate system. Furthermore, insome of these examples, P and Q have a value of ±1 as one component whenexpressed in the space-coordinate system. Conditions (i) and (ii)together imply that P, Q and S are always linearly independent. Thisimplies that they are also spanning, and so do form a basis. That theyform a non-degenerate basis is also demonstrated by the determinantnever being zero for a valid (i.e. nonzero) ray direction.

Some sort of error will be introduced into the projected quantities(e.g. through floating point evaluation) and the choice of basis vectorsis relevant to how this will affect the overall error of theintersection calculation. Given a general basis as defined above, thecomponents (p, q, s) of a projected vertex are given by the followingmatrix multiplications

${\begin{pmatrix}p \\q \\s\end{pmatrix} = {\begin{pmatrix}B & 0 & 0 \\0 & C & 0 \\0 & 0 & A\end{pmatrix}\begin{pmatrix}D_{z} & 0 & {- D_{x}} \\0 & D_{z} & {- D_{y}} \\D_{x} & D_{y} & D_{z}\end{pmatrix}\begin{pmatrix}v_{x} \\v_{y} \\v_{z}\end{pmatrix}}},$where (v_(x), v_(y), v_(z)) are the vertex components in world orinstance space after the ray origin has been subtracted and perhapsafter the axes have been permuted and mirrored such that D_(z)≥D_(x),D_(z)≥D_(y) and D_(z)≥0. The inverse mapping, which expresses thesevertex components in terms of their projected counterparts is given by

${\begin{pmatrix}v_{x} \\v_{y} \\v_{z}\end{pmatrix} = {{\frac{1}{D^{2}} \cdot \frac{1}{D_{z}} \cdot \begin{pmatrix}{D^{2} - D_{x}^{2}} & {{- D_{x}}D_{y}} & {D_{x}D_{z}} \\{{- D_{x}}D_{y}} & {D^{2} - D_{y}^{2}} & {D_{y}D_{z}} \\{{- D_{x}}D_{z}} & {{- D_{y}}D_{z}} & D_{z}^{2}\end{pmatrix}}\begin{pmatrix}\frac{1}{B} & 0 & 0 \\0 & \frac{1}{C} & 0 \\0 & 0 & \frac{1}{A}\end{pmatrix}\begin{pmatrix}p \\q \\s\end{pmatrix}}},$where D²=D_(x) ²+D_(y) ²+D_(z) ² is the square length of the ray. Theabove expression is useful because it illustrates how errors in thecalculated projected components map to errors in the original vertexlocations, thereby giving an impression of the error behaviour in aray-independent fashion (recall that a multitude of distinct rays may betested against a given primitive), in the sense that a direct comparisonmay be made between perturbed vertex locations as implied by one rayprojection versus another (assuming a shared ray origin).

Suppose, for example, that we compute p and q as p′ and q′ with relativeerror ε_(p) and ε_(q) (realistic for an accurate floating pointimplementation) such that

${\begin{pmatrix}p^{\prime} \\q^{\prime} \\s\end{pmatrix} - \begin{pmatrix}p \\q \\s\end{pmatrix}} = {\begin{pmatrix}\varepsilon_{p} & 0 & 0 \\0 & \varepsilon_{q} & 0 \\0 & 0 & 0\end{pmatrix}{\begin{pmatrix}p \\q \\s\end{pmatrix}.}}$

Note that, since s is not explicitly computed in this example, it may bedescribed, as above, with zero effective error. Now, using the earlierexpression for the inverse mapping, we may express this error in termsof (v′_(x), v′_(y), v′_(z)), the implied offset vertex, with

${\begin{pmatrix}v_{x}^{\prime} \\v_{y}^{\prime} \\v_{z}^{\prime}\end{pmatrix} - \begin{pmatrix}v_{x} \\v_{y} \\v_{z}\end{pmatrix}} = {{{\frac{1}{D^{2}} \cdot \frac{1}{D_{z}} \cdot \begin{pmatrix}{D^{2} - D_{x}^{2}} & {{- D_{x}}D_{y}} & {D_{x}D_{z}} \\{{- D_{x}}D_{y}} & {D^{2} - D_{y}^{2}} & {D_{y}D_{z}} \\{{- D_{x}}D_{z}} & {{- D_{y}}D_{z}} & D_{z}^{2}\end{pmatrix}}{\begin{pmatrix}\frac{1}{B} & 0 & 0 \\0 & \frac{1}{C} & 0 \\0 & 0 & \frac{1}{A}\end{pmatrix}\left\lbrack {\begin{pmatrix}p^{\prime} \\q^{\prime} \\s\end{pmatrix} - \begin{pmatrix}p \\q \\s\end{pmatrix}} \right\rbrack}} = {{\frac{1}{D^{2}} \cdot \frac{1}{D_{z}} \cdot \begin{pmatrix}{D^{2} - D_{x}^{2}} & {{- D_{x}}D_{y}} & {D_{x}D_{z}} \\{{- D_{x}}D_{y}} & {D^{2} - D_{y}^{2}} & {D_{y}D_{z}} \\{{- D_{x}}D_{z}} & {{- D_{y}}D_{z}} & D_{z}^{2}\end{pmatrix}}\begin{pmatrix}\frac{1}{B} & 0 & 0 \\0 & \frac{1}{C} & 0 \\0 & 0 & \frac{1}{A}\end{pmatrix}\begin{pmatrix}\varepsilon_{p} & 0 & 0 \\0 & \varepsilon_{q} & 0 \\0 & 0 & 0\end{pmatrix}{\begin{pmatrix}p \\q \\s\end{pmatrix}.}}}$

If we substitute our original projection expression for (p, q, s), thissimplifies to

${\begin{pmatrix}v_{x}^{\prime} \\v_{y}^{\prime} \\v_{z}^{\prime}\end{pmatrix} - \begin{pmatrix}v_{x} \\v_{y} \\v_{z}\end{pmatrix}} = {{{\frac{1}{D^{2}} \cdot \frac{1}{D_{z}} \cdot \text{ }\begin{pmatrix}{D^{2} - D_{x}^{2}} & {{- D_{x}}D_{y}} & {D_{x}D_{z}} \\{{- D_{x}}D_{y}} & {D^{2} - D_{y}^{2}} & {D_{y}D_{z}} \\{{- D_{x}}D_{z}} & {{- D_{y}}D_{z}} & D_{z}^{2}\end{pmatrix}}\begin{pmatrix}\varepsilon_{p} & 0 & 0 \\0 & \varepsilon_{q} & 0 \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}D_{z} & 0 & {- D_{x}} \\0 & D_{z} & {- D_{y}} \\D_{x} & D_{y} & D_{z}\end{pmatrix}\begin{pmatrix}v_{x} \\v_{y} \\v_{z}\end{pmatrix}} = {{\frac{1}{D^{2}}\left( {v_{x} - {\frac{D_{x}}{D_{z}}v_{z}}} \right)\begin{pmatrix}{D^{2} - D_{x}^{2}} \\{{- D_{x}}D_{y}} \\{{- D_{x}}D_{z}}\end{pmatrix}\varepsilon_{p}} + {\frac{1}{D^{2}}\left( {v_{y} - {\frac{D_{y}}{D_{z}}v_{z}}} \right)\begin{pmatrix}{{- D_{x}}D_{y}} \\{D^{2} - D_{y}^{2}} \\{{- D_{y}}D_{z}}\end{pmatrix}{\varepsilon_{q}.}}}}$

The first important thing to observe (in this example in which theprojected coordinates are computed with relative error) is that theimplied vertex error does not depend on the choice of values for A, B orC (assuming values of zero are correctly handled, as describedelsewhere). This allows us to decide these quantities on their relativemerits, without having to consider their impact on the error analysis.

Next, we can understand bounds on the error by writing the total squarerelative error ε² as

$\begin{matrix}{\varepsilon^{2} = {\frac{1}{v^{2}}\left\lbrack {\left( {v_{x}^{\prime} - v_{x}} \right)^{2} + \left( {v_{y}^{\prime} - v_{y}} \right)^{2} + \left( {v_{z}^{\prime} - v_{z}} \right)^{2}} \right\rbrack}} \\{= {{\frac{1}{v^{2}}\left( {v_{x} - {\frac{D_{x}}{D_{z}}v_{z}}} \right)^{2}\varepsilon_{p}^{2}} + {\frac{1}{v^{2}}\left( {v_{y} - {\frac{D_{y}}{D_{z}}v_{z}}} \right)^{2}\varepsilon_{q}^{2}} - {\frac{1}{v^{2}}\left\lbrack {{\frac{D_{x}}{D}\left( {v_{x} - {\frac{D_{x}}{D_{z}}v_{z}}} \right)\varepsilon_{p}} + {\frac{D_{y}}{D}\left( {v_{y} - {\frac{D_{y}}{D_{z}}v_{z}}} \right)\varepsilon_{q}}} \right\rbrack}^{2}}}\end{matrix}$where v²=v_(x) ²+v_(y) ²+v_(z) ² is the square length of the vertexrelative to the ray origin, for a vertex not at the ray origin. Now wecan see why it is important that D_(z) is the major axis; the aboveequation involves D_(x)/D_(z) and D_(y)/D_(z) terms that may becomeunbounded for finite ε_(x) and ε_(y), the errors on the computedprojected components, as D_(z) tends to zero.

Take for example the case where v_(z)≠0, ε_(p)≠0, ε_(q)=0, D_(x)=D_(y)and D_(z)/D_(x)→0. We have

$\begin{matrix}{\varepsilon^{2} = {\frac{1}{v^{2}}\left( {1 - \frac{D_{x}^{2}}{D^{2}}} \right)\left( {v_{x} - {\frac{D_{x}}{D_{z}}v_{z}}} \right)^{2}\varepsilon_{p}^{2}}} \\{= {{\left. \frac{1}{v^{2}}\left( \frac{D_{x}^{2} + D_{z}^{2}}{{2D_{x}^{2}} + D_{z}^{2}} \right)\left( {v_{x} - {\frac{D_{x}}{D_{z}}v_{z}}} \right)^{2}\varepsilon_{p}^{2} \right.\sim\frac{1}{2v^{2}}\left( {\frac{D_{x}}{D_{z}}v_{z}} \right)^{2}\varepsilon_{p}^{2}}\rightarrow\infty}}\end{matrix}$where in the first line we have used the substitution ε_(y)=0, in thesecond line we have used D_(x)=D_(y) and in the third line we have usedD_(z)/D_(x)→0 and the fact that v_(z)≠0. So we see that informationabout the vertex component v_(x) (and v_(y)) gets washed out in theprojection as D_(z)/D_(x)→0 as should be intuitively clear withreference to the matrix expression for (p, q, s). Of course, if D_(z) isthe major axis, the magnitude of D_(z)/D_(x) (and D_(z)/D_(y)) isbounded below by one so that the above pathology doesn't arise (and itis easy to construct an upper bound on the total error in terms of theerror bounds on the projected components).

A “reciprocal basis” is one where in the basis vectors,P=B(D_(z),0,−D_(x)) and Q=C(0,D_(z),−D_(y)), B is a function of

$\frac{1}{D_{x}}$and C is a function or

$\frac{1}{D_{y}}.$In other words, B is a simplified fraction with D_(x) in itsdenominator, and C is a simplified fraction with D_(y) in itsdenominator. For example, a reciprocal basis may have

${B = {{\frac{1}{D_{x}}{and}C} = \frac{1}{D_{y}}}},$such that

$P = {{\left( {\frac{D_{z}}{D_{x}},0,{- 1}} \right){and}Q} = {\left( {0,\frac{D_{z}}{D_{y}},{- 1}} \right).}}$As another example, a reciprocal basis may have

${B = {{\frac{1}{D_{x}D_{z}}{and}C} = \frac{1}{D_{y}D_{z}}}},$such that

$P = {{\left( {\frac{1}{D_{x}},0,{- \frac{1}{D_{z}}}} \right){and}Q} = {\left( {0,\frac{1}{D_{y}},{- \frac{1}{D_{z}}}} \right).}}$One advantage of using a reciprocal basis is that each reciprocal may beevaluated to some accuracy (usually less accurate than comparablefloating point operations) and then those evaluated values may betreated as if they were exact, in essence perturbing the direction ofthe ray direction for the sake of improving the accuracy of allintersection testers (improving consistency across the board). Anotheradvantage of using a reciprocal basis is that it provides moreflexibility in how the intersection distance, if required, iscalculated. In particular, if a point of intersection (relative to theray basis coordinate system) (p_(x), p_(y), p_(z)) is determined, forexample via barycentric interpolation, then the intersection parameter tmay be alternatively computed as

${\frac{1}{D_{x}}p_{x}},{\frac{1}{D_{y}}p_{y}{or}\frac{1}{D_{z}}{p_{z}.}}$This may provide superior error characteristics if, for example, thecomponent is selected corresponding to the axis for which a given convexpolygon's components have the smallest range. This is in contrast with anon-reciprocal basis for which it is less cost effective to perform acalculation involving the reciprocals

$\frac{1}{D_{x}}{and}\frac{1}{D_{y}}$without additional per ray storage. Pre-computing these additional termswould also potentially sacrifice some of the consistency in thedefinition of the ray since the computed reciprocals could not ingeneral be treated as exact. Since the reciprocal basis projectionoperation also involves products of vertex components of the form

${\frac{1}{D_{x}}v_{x}},{\frac{1}{D_{y}}v_{y}{and}\frac{1}{D_{z}}v_{z}},$these intermediate products may be retained so that the intersectionparameter t is obtained directly from barycentric interpolation withoutthe need for any further scaling (described in more detail later in thedocument).

Table 1 below shows eight examples (denoted A to H) of different rayformats and basis vectors (up to sign) which may be used in differentexamples. The phrase “up to sign” here means that P and Q can be scaledby any unit value, i.e. restricted to the set {−1. +1}. Table 1 alsoshows the number of multiply and reciprocal operations that would beperformed per ray for the different ray formats, the number of multiplyoperations that would be performed per vertex for the projection into 2Dray space with its corresponding basis, and the reciprocal and multiplyoperations that would be performed per primitive for the intersectiondistance calculation.

TABLE 1 Example ray formats and basis vectors Ray Distance DistanceImplicit Basis Ray Muls Recips Muls (per Recips Muls (per # Ray Formatvectors (per Ray) (per Ray) Vertex) (per Prim.) Prim.) A (D_(x), D_(y),D_(z)) P = (D_(z), 0, −D_(x)) 0 0 4 1 1 B$\left( {D_{x},D_{y},\frac{1}{D_{z}}} \right)$ Q = (0, D_(z), −D_(y)) 01 4 0 1 C $\left( {\frac{1}{D_{x}},\frac{1}{D_{y}},D_{z}} \right)$$P = \left( {\frac{1}{D_{x}},0,{- \frac{1}{D_{z}}}} \right)$ 0 2 3 1 0 D$\left( {\frac{1}{D_{x}},\frac{1}{D_{y}},\frac{1}{D_{z}}} \right)$$Q = \left( {0,\frac{1}{D_{y}},{- \frac{1}{D_{z}}}} \right)$ 0 3 3 0 0 E$\left( {\frac{D_{x}}{D_{z}},\frac{D_{y}}{D_{z}},D_{z}} \right)$$P = \left( {1,0,{- \frac{D_{x}}{D_{z}}}} \right)$ 2 1 2 1 1 F$\left( {\frac{D_{x}}{D_{z}},\frac{D_{y}}{D_{z}},\frac{1}{D_{z}}} \right)$$Q = \left( {0,1,{- \frac{D_{y}}{D_{z}}}} \right)$ 2 1 2 0 1 G$\left( {\frac{D_{z}}{D_{x}},\frac{D_{z}}{D_{y}},D_{z}} \right)$$P = \left( {\frac{D_{z}}{D_{x}},0,{- 1}} \right)$ 2 2 2 1 1 H$\left( {\frac{D_{z}}{D_{x}},\frac{D_{z}}{D_{y}},\frac{1}{D_{z}}} \right)$$Q = \left( {0,\frac{D_{z}}{D_{y}},{- 1}} \right)$ 2 3 2 0 1

The different options for the ray formats and basis vectors can becompared with each other based on the total cost of all the floatingpoint operations (FLOPs) from both the ray format encoding, the basisprojection and the distance calculation. When the pre-processing of aray into a given format (e.g. one of the options shown as A to H inTable 1) is decoupled from the polygon intersection testing andperformed once per ray, and given that a single ray is tested againstmultiple vertices, then offloading FLOP costs from the basis projectiononto the ray format encoding to minimise the former tends to bebeneficial. When a single ray is tested against multiple primitives,then offloading FLOP costs from the distance calculation onto the rayformat encoding to minimise the former is also beneficial.

Options B, D, F and H are generally all improvements over A, C, E and Grespectively, as they shift the reciprocal from the (per primitive)distance calculation to the (per ray) ray format encoding.

Option B (and option A) is the default ray format/basis which is themost intuitive way to perform the intersection testing but is not themost efficient in terms of the number of floating point operations thatneed to be performed. Option F (and option E) reduces the number ofmultiply operations that are performed per vertex in the basisprojection by two relative to option B (and option A).

Options C, D, G and H use reciprocal bases. Option D (and option C)reduces the number of multiply operations that are performed per vertexin the basis projection by one, and reduces the number of multiplyoperations that are performed per primitive in the distance calculationby one, relative to option B (and option A).

Option H (and option G) reduces the number of multiply operations thatare performed per vertex in the basis projection by two relative tooption B (and option A).

Using the ray-coordinate system with basis vectors as described in theexamples above can simplify some of the processing involved inintersection testing. In particular, if a basis vector has a 0 as acomponent value then a multiply and/or add operation (e.g. as used forperforming a dot product or a cross product) involving that basis vectorwill not include a multiply and/or an add operation for the componentwhich is 0, thereby reducing the number and/or complexity of operationsthat need to be performed. Similarly, if a basis vector has ±1 as acomponent value then a multiply and/or add operation (e.g. as used forperforming a dot product or a cross product) involving that basis vectorwill not include a multiply operation for the component which is ±1,thereby reducing the number and/or complexity of operations that need tobe performed. Reducing the number of operations that are performed willtend to reduce the latency and power consumption of the intersectiontesting module 108 and, additionally, reducing the complexity ofoperations that are performed will tend to increase the accuracy.Furthermore, when the intersection testing module 108 is implemented infixed function circuitry then reducing the number and/or complexity ofoperations that are performed will tend to reduce the size (i.e. thesilicon area) of the intersection testing module 108. For example, thefixed function circuitry may comprise one or more multiply-and-addcomponents (e.g. a fused singly rounded multiply-add unit) forperforming multiplications and additions using the second and thirdbasis vectors of the ray-coordinate system. In a singly rounded floatingpoint implementation (i.e. rounding is applied to the exact final resultto ensure a representable value is output), this is more accurate andtypically smaller in area than a sum of products, multiply rounded (bothproducts and sum). Its area may even be comparable to a less accuratemultiply rounded implementation, which is less likely to be the case formore complex operations. Floating point add and multiply operations(unlike the real numbers they approximate) are not associative. Reducingthe operations, which reduces the possible orderings, thus may haveother benefits in terms of consistency of results.

In the main examples described herein the ray data for the rays has n=3coefficients. However, in other examples, the ray data could have morethan three coefficients. The construction of the ray coordinate systemdescribed can be applied to any dimension n≥2. For example, n basisvectors of the ray coordinate system can be defined where a first of thebasis vectors of the ray coordinate system is aligned with the raydirection, and there are (n−1) other basis vectors which are orthogonalto the first basis vector. If n>2 then no pairing of the (n−1) otherbasis vectors are parallel. Furthermore, the (n−1) other basis vectorshave zeros for (n−2) components when expressed in the space-coordinatesystem. The n basis vectors of the ray coordinate system form a basis ofn-dimensional space (they are linearly independent and hence spanning).For example, in a four dimensional space, the ray direction vector maybe D=(D_(w),D_(x),D_(y),D_(z)), and the four basis vectors of the raycoordinate system may be: S=A(D_(w),D_(x),D_(y),D_(z)),P=B(D_(z),0,0,−D_(w)) and Q=C(0,D_(z),0,−D_(z)), R=D(0,0,D_(z),−D_(y)),where in this example, the selective permutation of the axes ensuresthat D, is the major component of the ray direction vector, i.e.|D_(z)|≥|D_(w)|, |D_(z)|≥|D_(x)| and |D_(z)|≥|D_(y)|. In this example,A, B, C and D are nonzero scalars, and they may be chosen to make one ofthe coefficients unital, e.g. as described above in relation to thethree dimensional examples. In this example, the four dimensions maycorrespond to three spatial dimensions and one temporal dimension. Inthe 4D case, an expression for the sign of the determinant (similar toexpression (1) given above for the 3D case) may be given as:sgn(A) XOR sgn(B) XOR sgn(C) XOR sgn(D) XOR sgn(σ)  (2)

As an example, the permutation sigma may reverse the mapping sign if themajor component of the ray direction vector is the W or the Y axis, butnot if the major component of the ray direction vector is the X or the Zaxis. In particular, in this example, permuting the components so thatthe major component is in the final position may involve a four cyclefor a major axis of W and a transposition for a major axis of Y, in bothcases flipping the sign. The sign of D_(z) does not factor in expression(2) because n is even (i.e. n=4).

Returning to the 3D case, in some examples, the ray and the convexpolygon are transformed from the space-coordinate system into theray-coordinate system, wherein the intersection testing is performed inthe ray-coordinate system. Testing in the ray-coordinate system could beperformed by determining if the ray intersects with the convex polygondefined by its vertices as defined in the ray-coordinate system.However, in some other examples, a full transformation of the ray andthe convex polygon into the ray-coordinate system does not need to beperformed. In these other examples, the polygons can be projected onto apair of axes corresponding to the second and third basis vectors of theray coordinate system, but components of the polygons along the raydirection (i.e. in the direction of the first basis vector of the raycoordinate system) might not be determined. It is noted that definingthe ray in the ray-coordinate system is trivial because its origin is atthe origin of the ray-coordinate system and its direction is parallel tothe S axis, such that its component values along the P and Q axes of theray-coordinate system are zero.

In the examples described above, the axes are selectively permuted toensure that D_(z) is the major component of the ray direction vector.Furthermore, the axes may be selectively reversed (or “reflected”) toensure that all of the components of the ray direction vector arenon-negative, i.e. D_(z)>0, D_(x)≥0 and D_(y)≥0. The selective reversingof the axes causes the ray direction vector to point into the octant ofthe space-coordinate system which has positive values for x, y and z.Since D_(z) is the major component of the ray direction vector thismeans that

$\frac{D_{x}}{D_{z}}{and}\frac{D_{y}}{D_{z}}$are in a range from 0 to 1 and it means that

$\frac{D_{z}}{D_{x}}{and}\frac{D_{z}}{D_{y}}$are in an interval from 1 to positive infinity. After the selectivepermutation and selective reversing of the axes, D_(z)≥D_(x)≥0 andD_(z)≥D_(y)≥0.

Returning to the sign of the change of basis, forcing the signs ofD_(x), D_(y) and D_(z) to be nonnegative means that any appearance ofthem in the determinant, sgn(A), sgn(B) or sgn(C) is forced to 1,simplifying the post sign modification. However, as the vectorcoordinates must be reversed/reflected to match the reversal/reflectionof the ray coordinates, the sign must be post-modified by sgn(Dx),sgn(Dy) and sgn(Dz). So for the particular choices of A, B and C givenin the example above, this results in the same sign modificationexpression.

A mapping indication of the sign of the change-of-basis mapping can bestored, wherein the mapping indication indicates whether the mappingfrom the space-coordinate system to the ray-coordinate system affectsthe perceived polygon orientation. The mapping indication can be usedduring intersection testing of the ray with a polygon to ensure thecorrect orientation (perceived by the ray in the originalspace-coordinate system) is output.

FIG. 3 is a flow chart for a method of performing intersection testingfor a ray with respect to a plurality of convex polygons (e.g.triangles), performed by the polygon intersection testing unit(s) 114 ofthe intersection testing module 108. Prior to the intersection testingof the ray with respect to the polygons performed by the polygonintersection testing unit(s) 114, the ray may be tested by the boxintersection testing unit(s) for intersection, with boxes representingnodes of an acceleration structure, wherein the ray is scheduled forintersection with a polygon if the ray is determined to intersect a leafnode of the acceleration structure which references the polygon.

In step S302 data defining the ray 202 and data for the verticesdefining the convex polygons 204 and 206 are obtained at theintersection testing module 108. In particular, data defining thecomponents of the ray origin and the ray direction in thespace-coordinate system are obtained. The data defining the ray originmay be the three components of the ray origin position in thespace-coordinate system, O_(x), O_(y) and O_(z). In the example shown inFIG. 3 , the data defining the ray direction comprises the threecomponents of the ray direction in the space-coordinate system, D_(x),D_(y) and D_(z). Alternatively, as described above, some differentvalues defining the ray direction may have been pre-computed and storedin a store, such that in step S302 the pre-computed values may be read.For example, three values may be read to define the ray direction data,and these three values may be

$\frac{D_{x}}{D_{z}},{\frac{D_{y}}{D_{z}}{and}{\frac{1}{D_{z}}.}}$In other examples, afferent pre-computed values may be read to definethe ray direction, e.g. values of

$\frac{D_{z}}{D_{x}},\frac{D_{z}}{D_{y}}$and D_(z) may be read. As another example, values of

$\frac{D_{z}}{D_{x}},{\frac{D_{z}}{D_{y}}{and}\frac{1}{D_{z}}}$may be read. As another example, values of

$\frac{1}{D_{x}},{\frac{1}{D_{y}}{and}\frac{1}{D_{z}}}$may be read. In other examples, other values may be pre-computed andread which can be used to define the ray direction. The roles of thesethree values are distinct, and they are not always all needed. A firstand key purpose of a triangle tester is an intersection determination.This uses the first two values, e.g. either

$\frac{D_{x}}{D_{z}}{and}\frac{D_{y}}{D_{z}}{or}\frac{D_{z}}{D_{x}}{and}\frac{D_{z}}{D_{y}}$or similar, possibly with some signs depending on the choice of B and C.These are used to generate P and Q against which the primitive verticesare projected. The barycentric coordinates of an intersection point canbe derived from the intersection determination without additional input.A second purpose of a triangle tester is an orientation determination(e.g. for back-face culling). This additionally uses the sign of thethird value

${{sgn}\left( \frac{1}{D_{z}} \right)} = {{sgn}\left( D_{z} \right)}$to ensure correct results, as per the determinant form. A third andfinal purpose of a triangle tester is an intersection distancecalculation. This additionally uses the magnitude of the third value,i.e. either

$\frac{1}{❘D_{z}❘}$or |D_(z)|, to rescale the distance value correctly. If the thirdpurpose is not needed (e.g. when the ray has infinite range or t_(min)and t_(max) have been pre-scaled accordingly), then

$\frac{1}{❘D_{z}❘}$or |D_(z)| may not be required as input. If the second purpose is notneeded (e.g. when no distinction is made between hitting either side ofthe primitive), then sgn(D_(z)) may not be required. Alternatively, anintersection distance calculation may use any of

$\frac{1}{D_{x}},{\frac{1}{D_{y}}{or}\frac{1}{D_{z}}}$in conjunction with the corresponding component of the intersectionpoint to determine the intersection distance.

In the example described with reference to FIG. 3 , data defining the x,y and z components of the positions of the vertices defining the convexpolygons 204 and 206 is read in step S302.

In step S304 the x, y and z components of the data defining the ray andthe vertices defining the convex polygons are selectively permutedand/or reversed by the intersection testing module 108 (e.g. by the rayrescaling unit 116), such that D_(z)≥D_(x)≥0 and D_(z)≥D_(y)≥0. Thepermutation of the axes may be thought of as rearranging the axes. Inparticular, a permutation of the axes comprises either a rotation ofthree axes, a transposition of two axes or the identity (i.e. notchanging the axes). It is noted that a permutation involving atransposition of two axes will alter the handedness of the coordinatesystem, whereas a permutation that does not involve a transposition oftwo axes will not alter the handedness of the coordinate system. Thepermutation of the axes is performed so that the major component of theray direction is D_(z) (i.e. ensuring that |D_(z)|≥|D_(x)| and|D_(z)|≥|D_(y)|). For example, if the original z component of the raydirection vector has a larger magnitude than the original x and ycomponents of the ray direction vector then no permutation is used(which may be thought of as the permutation using the identityoperation); if the original x component of the ray direction vector hasa larger magnitude than the original y and z components of the raydirection vector then the permutation may comprise a rotation of thethree axes such that the x components become the z components, the zcomponents become the y components, and the y components become the xcomponents; and if the original y component of the ray direction vectorhas a larger magnitude than the original x and z components of the raydirection vector then the permutation comprises a transposition of the yand z axes such that the y components become the z components, and the zcomponents become the y components (and the x components stay as the xcomponents). It would be possible to just perform the selectivepermutation in step S304 (i.e. not perform the selective reversing), butin the method described with reference to FIG. 3 , the selectivereversing of zero, one or more of the axes is performed so that the raydirection is into the octant with positive x, y and z. The “reversing”of the axes, may be referred to as “reflecting”, “inverting” or“negating”, and may involve changing the sign of all of the componentvalues in the dimension along the axis in question. The data definingthe ray comprises the ray origin and the ray direction, and the datadefining the convex polygon comprises three or more vertex positions. Itis noted that reversing an odd number of the axes reverses the perceivedorientation of the polygons.

In step S306, the intersection testing module 108 projects the verticesof the convex polygons onto a pair of axes orthogonal to the raydirection, wherein the origin of the pair of axes corresponds with theray origin. This projection of the vertices of the convex polygons ontoa pair of axes orthogonal to the ray direction may comprise transformingthe vertices of the convex polygons into the ray coordinate systemdescribed above. As described above, the ray-coordinate system has anorigin at the ray origin, so step S306 may involve subtracting therespective components of the ray origin (O_(x), O_(y) and O_(z)) fromrespective components of the data defining the positions of the verticesdefining the polygons, to thereby determine components of the positionsof the vertices defining the polygons relative to the ray origin. Asdescribed above, the ray-coordinate system has three basis vectors,wherein a first of the basis vectors is aligned with the ray direction;and wherein a second and a third of the basis vectors are respectivelyaligned with said pair of axes orthogonal to the ray direction, whereinthe second and third basis vectors: (i) are not parallel with eachother, and (ii) have a 0 as one component when expressed in thespace-coordinate system. As mentioned above, in some examples, thesecond and third basis vectors may have ±1 as one component whenexpressed in the space-coordinate system.

The projection can take the form of a dot product. The projection of a3D vertex v=(v_(x),v_(y),v_(z)) to a 2D ray-space coordinate (p, q),with 3D basis vectors P=(P_(x),P_(y),P_(z)) and Q=(Q_(z),Q_(y),Q_(z)) isas follows:v→p=P·v=P _(x) v _(x) +P _(y) v _(y) +P _(z) v _(z)v→q=Q·v=Q _(x) v _(x) +Q _(y) v _(y) +Q _(z) v _(z)

With the examples of the basis described herein, P_(y)=0 and Q_(x)=0,such that:p=P _(x) v _(x) +P _(z) v _(z)q=Q _(y) v _(y) +Q _(z) v _(z)

This may be implemented by separate multiplications and an addition, oralternatively as a (singly rounded) fused DP2 (2D dot product), which inthis latter case would ensure that the computed components have finiterelative error with the properties discussed earlier.

Furthermore, with some of the examples of the basis described hereinP_(x) or P_(z) may be equal to ±1 and Q_(y) or Q_(z) may be equal to ±1,respectively, such that either:p=±v _(x) +P _(z) v _(z)q=±v _(y) +Q _(z) v _(z)Or:p=P _(x) v _(x) ±v _(z)q=Q _(y) v _(y) ±v _(z)

Either of these may be implemented by a separate multiplication andaddition, or alternatively as a (singly rounded) FMA (fused multiplyadd), which in this latter case would ensure that the computedcomponents have finite relative error with the properties discussedearlier.

As mentioned above, a mapping indication is stored to indicate whethermapping from the space-coordinate system to the ray-coordinate system(i.e. steps S304 and S306) has altered the perceived orientation of thepolygons. If the perceived orientation of the polygons has changed thenthis is taken into account in the intersection process, e.g. in stepS318, in order to ensure that the mapping from the space-coordinatesystem to the ray-coordinate system in steps S304 and S306 does notprevent the intersection testing module from correctly determiningwhether a ray intersects a polygon.

FIG. 6 illustrates the two polygons 204 and 206 shown in FIG. 2 whenthey have been projected onto the pair of axes aligned with the basisvectors P and Q (i.e. the second and third basis vectors of the raycoordinate system). In this projection, the ray is at the origin anddirected into the page, i.e. the components of the ray direction vectoralong the P and Q axes are zero. Therefore, the intersection position208 representing the intersection of the ray with the polygon 204 is atthe origin in FIG. 6 . Figure six indicates that the orientations of thetwo polygons 204 and 206 both appear as clockwise in FIG. 2 , asdetermined by the winding order of the vertices in the ordered sets ofvertices defining the polygons. In FIG. 6 , the ray coordinate systembasis vectors P and Q are shown at a right angle to each other eventhough they may not be orthogonal in the space coordinate system.However, it is clear that this visualisation does not affect theintersection determination for a convex polygon (i.e., whether it coversthe origin or not).

In step S308 a parameter, k, is set to zero. The parameter, k, is usedto indicate a particular convex polygon.

In step S310 a parameter, e, is set to zero. The parameter, e, is usedto indicate a particular edge of a convex polygon. It is noted that thepolygon's winding order (i.e., the order of its vertices) specifies adirection for each of its edges, such that every edge given below isimplicitly a directed edge. In the example in which the polygons aretriangles then the maximum value (e_(max)) for the parameter, e, is 2.In other words the parameter, e, can take values of 0, 1 or 2 toindicate the three edges of a triangle. In the example in which thepolygons have n edges then the maximum value (e_(max)) for theparameter, e, is n−1. Each edge of a convex polygon is defined by two ofthe projected vertices as an ordered pair, indicating the direction ofthe edge. For example polygon 204 shown in FIG. 6 has a first edgedefined by the projected vertices v₀ and v₁, i.e., directed from v₀ tov₁, a second edge defined by the projected vertices v₁ and v₂, i.e.,directed from v₁ to v₂, and a third edge defined by the projectedvertices v₂ and v₀, i.e., directed from v₂ to v₀. Similarly, polygon 206shown in FIG. 6 has a first edge defined by the projected vertices v₁and v₃, i.e., directed from v₁ to v₃, a second edge defined by theprojected vertices v₃ and v₂, i.e., directed from v₃ to v₂, and a thirdedge defined by the projected vertices v₂ and v₁, i.e., directed from v₂to v₁. The edge between projected vertices v₁ and v₂ is shared betweenpolygons 204 and 206, albeit in opposite directions indicating thatpolygon 204 and 206 are specified with consistent winding orders. Insome examples, the method can iterate over each unique edge just theonce, rather than iterating over n edges for each convex polygonseparately, to take advantage of the sharing of some edges.

In step S312 the polygon testing unit(s) 114 of the intersection testingmodule 108 determines a parameter, w, indicative of which side of thee^(th) edge of the k^(th) polygon the ray passes on. For example, w maybe a signed parameter which is determined using a function (which isreferred to herein as a “2D cross product”), f(v_(i),v_(j)), of thepositions of the two projected vertices defining the e^(th) edge of thek^(th) polygon. The 2D cross product, f(v_(i),v_(j)), of the positionsof two projected vertices, v_(i) and v_(j), defining an edge, is definedas f(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j), where p_(i) and q_(i) arecomponents of the projected vertex v_(i) along the respective axes ofthe pair of axes, and where p_(j) and q_(j) are components of theprojected vertex v_(j) along the respective axes of the pair of axes. Inother examples, other functions of the positions of the two projectedvertices defining the edge may be used to determine the parameter w. Forexample, the function may return the result of a comparison ofp_(i)q_(j) and q_(i)p_(j), which would avoid performing a subtractionoperation, but would not provide a magnitude for use in determiningbarycentric coordinates. The sign of w for the e^(th) edge of the k^(th)polygon indicates whether the ray passes on the left or the right ofthat edge, from the edge's perspective (directed from v_(i) to v_(j)).Passing on the “left” side of the edge corresponds to an anticlockwiserotation from the first endpoint to the second endpoint relative to theorigin. Passing on the “right” side of the edge corresponds to aclockwise rotation from the first endpoint to the second endpointrelative to the origin. With the form of the “2D cross product” givenabove as f=(v_(i),v_(j))=p_(i)q_(j)−p_(i)p_(j), and with a first axispointing rightwards and a second axis pointing upwards,left/anticlockwise corresponds to a positive result of f andright/clockwise corresponds to a negative result of f. In the IEEEfloating point specification, a negative value has a sign bit of 1,whereas a positive value has a sign bit of 0. Assuming a left-handedsystem (as per FIG. 2 ), and aligning the first axis P and second axis Qwith right and up respectively in projected space, means the third axisS (the ray direction) points into the page, giving the correctedperceived orientation. In a right-handed system the third axis S pointsout of the page, giving the wrong perceived orientation. To summarise wehave the following correspondences: for a left-handed system: (i) leftside↔anticlockwise↔positive↔0 for the sign bit of f; and (ii) rightside↔clockwise↔negative↔1 for the sign bit of f. For a right-handedsystem: (i) left side→anticlockwise→negative→1 for the sign bit of f;and (ii) right side→clockwise→positive→0 for the sign bit of f. In someexamples, “anticlockwise” is encoded as a 1, and “clockwise” is encodedas a 0. Therefore, in a right-handed system the sign bit of w gives thecorrect flag state; where as in a left-handed system the sign bit of wgives the opposite flag state, so the orientation result ispost-inverted. To achieve this, a winding flag may be supplied to theuser, to set when a left-handed space coordinate system is required, toindicate that all edges should be treated as pointing in the oppositedirection.

The parameter w is determined using the 2D cross product in differentways in different implementations. Two examples are described in detailbelow with reference to FIGS. 4 and 5 respectively. For example, the twoprojected vertices v_(i) and v_(j), defining an edge, may be provided tothe 2D cross product, f(v_(i),v_(j)), in the order given by the directededge, which is inherited from the ordered set of vertices defining theconvex polygon (i.e., its winding order). For example, the 2D crossproduct for the shared edge of polygons 204 and 206 will be given asf(v₁,v₂)=p₁q₂−q₁p₂ for the edge of the polygon 204, and will be given asf(v₂,v₁)=p₂q₁−q₂p₁ for the edge of the polygon 206.

The function f(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j), referred to herein asa “2D cross product”, is a 2D analogy to the 3D triple product: itcalculates the signed area of a parallelogram defined by a first edgebetween the origin and (p₀, g₀) and a second edge between the origin and(p₁, q₁). This is equivalent to generating the determinant of the matrixwith rows (p_(i), q_(i)) for i=0,1. It is also equivalent to consideringthe endpoints as complex numbers, taking the complex conjugate of thefirst endpoint, multiplying this with the second endpoint and looking atthe imaginary component. Considering all cases geometrically, the sineof the angle between the two endpoints and the origin (i.e. intersectionpoint) is found, scaled by the distances of the two endpoints from theorigin. Sine is an odd function, so we have a signed function, i.e. weget a positive result if we rotate anticlockwise from the first endpointto the second endpoint, and a negative result if we rotate clockwisefrom the first endpoint to the second endpoint (with standard axes).

After parameter w is determined using the 2D cross product for thee^(th) edge of the k^(th) polygon in step S312, then in step S314 thepolygon testing unit(s) 114 of the intersection testing module 108determines whether e=e_(max). In other words, it determines whether thew parameter has been determined for all of the edges of the k^(th)polygon. If not, then the method passes to step S316 in which theparameter, e, is incremented, and then the method passes back to stepS312 so that the parameter w can be determined for the next edge of thek^(th) polygon. If it is determined in step S314 that the w parameterhas been determined for all of the edges of the k^(th) polygon (i.e. ife=e_(max)) then the method passes to step S318.

In step S318 the polygon intersection testing unit(s) 114 of theintersection testing module 108 determines whether the ray intersectsthe k^(th) polygon based on the w parameters determined for the edges ofthat polygon. For example, the determination as to whether the rayintersects the k^(th) polygon may be based on the signs of the wparameters determined for the edges of the k^(th) polygon. In anexample, if the w parameters determined for the edges of the k^(th)polygon all have the same sign then it is determined that the rayintersects the k^(th) polygon; whereas if it is not the case that the wparameters determined for the edges of the k^(th) polygon all have thesame sign then it is determined that the ray does not intersect theconvex polygon. In this way, step S318 comprises using the parametersdetermined for the edges of the convex polygon to determine whether theray passes on the inside of the edges of the convex polygon, wherein itis determined that the ray intersects the convex polygon if it isdetermined that the ray passes on the inside of all of the edges of theconvex polygon, and wherein it is determined that the ray does notintersect the convex polygon if it is determined that the ray passes onthe outside of one or more of the edges of the convex polygon.

In some examples, as soon as an edge test indicates that the ray passeson the outside of an edge of a polygon, then a determination can be madethat the ray does not intersect the polygon, without necessarilyperforming the edge tests for all of the edges of the polygon. Forexample, step S314 could be modified to also identify, after we haveprocessed at least 2 edges, if we have one or more determined wparameters that are strictly positive and one or more w parameters thatare strictly negative for edges of a polygon, and in that case thepolygon may be marked as a “miss” without determining w parameters forany remaining edges of the polygon.

Furthermore, for an intersection to be determined, as well as all nsigns of w matching, the signs of w might need to match somepredetermined value (e.g., when back-face culling), and if they do notmatch the predetermined value then the polygon can be discarded. Forexample, if culling is enabled for clockwise/anticlockwise primitives orfront/back-facing primitives (e.g., back-face culling), then as soon asa single edge test result is determined then it might be possible todiscard the primitive. For example, if anticlockwise primitives arebeing culled then any primitive with an intersection point on the leftof an edge can be discarded, likewise if clockwise primitives are beingculled then any primitive with an intersection point on the right of anedge can be discarded. Furthermore, many edges may be shared (perhapswith reversed directions) within a single batch of convex polygons, andin this situation each unique edge may be tested once and the resultsfor a shared edge can be duplicated (perhaps with reversed sign) acrossboth polygons sharing the edge.

In some examples, the presence of one or more w parameters with zeromagnitude may be treated as a special case. For example, if the wparameters for all of the edges of the polygon have non-zero magnitude,it may be determined that the ray intersects the k^(th) polygon only ifall edges have matching signs for their w parameters (similar to asdescribed above), but if, one or more w parameters have zero magnitudethen an intersection is determined only if all of the zero magnitude wparameters (for edges of nonzero projected length) have a particularsign (e.g. a positive sign, or in other examples, a negative sign) andif all remaining edges with nonzero w parameters have mutuallyconsistent sign. In these examples, the signs of the one or more wparameters which have zero magnitude do not necessarily have to matchthe signs of the one or more w parameters which have non-zero magnitudeto determine that the ray intersects the polygon. In this way, thedetermination of whether the ray intersects the convex polygon (in stepS318) comprises: (i) determining that the signed parameter, w, for anyedge of the convex polygon which has zero magnitude has a particularsign (e.g. positive in some examples, and negative in other examples),and (ii) determining that the signed parameters, w, for edges of theconvex polygon which have non-zero magnitude have signs that areconsistent with each other (albeit not necessarily consistent with thesigns of the w parameters which have zero magnitude). Furthermore, insome examples, it may be determined that the ray does not intersect thek^(th) polygon if all edges generate w parameters with zero magnitude,indicating degeneracy. In other words, in order to omit degeneratepolygons, if the 2D cross products for all of the edges of the convexpolygon have a magnitude of zero then it can be determined that the raydoes not intersect the convex polygon. If culling is enabled, primitivesmay be discarded based on the sign of the edges with nonzero wparameters, as described in the previous paragraph. Furthermore, asdescribed and in addition to the previous paragraph, it is not necessaryto determine all n w parameters once any pair of edges with nonzero wparameters is determined to have conflicting signs or, in some examples,once a w parameter is determined to have zero magnitude. In this lattercase, it may be determined that intersection does not occur if the wparameter (for an edge of nonzero projected length) does not have thenecessary sign. Alternatively, for a nondegenerate strictly convexpolygon, it may be determined that intersection does occur if preciselytwo w parameters (for edges of nonzero projected length) both have thenecessary sign (guaranteeing sign consistency of the remaining edgeswith nonzero w parameters), since the magnitudes of at most two wparameters (for edges of nonzero projected length) may be zero (one zerow parameter indicates edge intersection and two zero w parametersindicate vertex intersection). In the “sign-equality test”, a pair ofequal signs of zero parameters (for edges of nonzero projected length)likewise determines an intersection, but this property often may not beexploited as edges with zero w parameters are generally notdistinguished in this intersection test. This latter case assumesculling has not been enabled, as this may require checking the sign ofone of the edges with nonzero w parameters.

In some examples, the signed parameter, w, for an edge equals the 2Dcross product for the edge. In other examples, the signed parameter, w,equals the 2D cross product multiplied by a positive constant (e.g., ½).In other examples, the signed parameter, w, equals the 2D cross productmultiplied by a negative constant (e.g., −1), but in this case allorientation determinations are reversed. In the example shown in FIG. 6, the 2D cross product for the first edge of polygon 204 defined byvertices v₀ and v₁ is given by f(v₀,v₁)=p₀q₁−q₀p₁, and it will beappreciated by considering FIG. 6 that this is a negative nonzero value.For a polygon with a clockwise orientation as viewed given its winding(as indicated in FIG. 6 ) a negative nonzero value for the 2D crossproduct for an edge indicates that the ray may pass on the inside of theconvex polygon, as the ray passes on the right of the edge, from the(directed) edge's perspective, whereas a positive non-zero value for the2D cross product for an edge indicates that the ray does pass on theoutside of the convex polygon, as the ray passes on the left of theedge, from the (directed) edge's perspective. Conversely, for a polygonwith an anticlockwise orientation as viewed (not shown in FIG. 6 ) anegative non-zero value for the 2D cross product for an edge indicatesthat the ray does pass on the outside of the convex polygon, as the raypasses on the right of the edge, from the (directed) edge's perspective,whereas a positive non-zero value for the 2D cross product for an edgeindicates that the ray may pass on the inside of the convex polygon, asthe ray passes on the left of the edge, from the (directed) edge'sperspective. For a polygon of either orientation as viewed, a zero valuefor the 2D cross product for an edge indicates that ray intersectsexactly with the edge, i.e. with the boundary of the convex polygon. The2D cross product for the second edge of polygon 204 defined by verticesv₁ and v₂ is given by f(v₁,v₂)=p₁q₂−q₁p₂, and it will be appreciated byconsidering FIG. 6 that this is a negative value, thereby indicatingthat the ray may pass on the inside of the convex polygon, as the raypasses on the right of the second edge. The 2D cross product for thethird edge of polygon 204 defined by vertices v₂ and v₀ is given byf(v₂, v₀)=p₂q₀−q₂p₀, and it will be appreciated by considering FIG. 6that this is a negative value, thereby indicating that the ray may passon the inside of the convex polygon, as the ray passes on the right ofthe third edge. Having checked all three edges of the convex polygon 204the w parameters, which are based on the 2D cross products, have thesame sign (in particular they are all negative, indicating a clockwiseorientation as viewed), so the ray is determined to pass on the insideof and intersect the polygon 204. It can be seen that the intersectionpoint 208 is within the polygon 204.

The 2D cross product for the first edge of polygon 206 defined byvertices v₁ and v₃ is given by f(v₁,v₃)=p₁q₃−q₁p₃, and it will beappreciated by considering FIG. 6 that this is a negative value. Polygon206 has a clockwise orientation as viewed given its winding, so the raymay pass on the inside of the convex polygon, as the ray passes on theright of the first edge of polygon 206 defined by vertices v₁ and v₃.The 2D cross product for the second edge of polygon 206 defined byvertices v₃ and v₂ is given by f(v₃,v₂)=p₃q₂−q₃p₂, and it will beappreciated by considering FIG. 6 that this is a negative value, therebyindicating that the ray may pass on the inside of the convex polygon, asthe ray passes on the right of the second edge. The 2D cross product forthe third edge of polygon 206 defined by vertices v₂ and v₁ is given byf(v₂,v₁)=p₂q₁−q₂p₁, and it will be appreciated by considering FIG. 6that this is a positive value, thereby indicating that the ray does passon the outside of the convex polygon, as the ray passes on the left ofthe third edge. Therefore, it is not the case that the w parameters,which are based on the 2D cross products, for all the edges of polygon206 have the same sign. In particular, the w parameters, which are basedon the 2D cross products, for the first and second edges of polygon 206are negative, whilst the w parameter, which is based on the 2D crossproduct, for the third edge of polygon 206 is positive. Therefore, theray is determined to not intersect the polygon 206, nor can a consistentorientation be established. It can be seen that the intersection point208 is outside the polygon 206. It is noted that for a valid convexpolygon, it is not possible for the ray to pass on the outside of allthe edges of the polygon (e.g. for a polygon with a clockwiseorientation as viewed, it is not possible for the ray to pass on theleft of all of the edges of the polygon), so if the w parameters, whichare based on the 2D cross products, for all the edges of the polygonhave the same sign then this indicates that the ray passes on the insideof all the edges, such that the ray is determined to intersect thepolygon (e.g. if the w parameters for all the edges of a polygon with aclockwise orientation have the same sign then this indicates that theray passes on the right of all the edges). It is also noted that for avalid convex polygon and an anticlockwise orientation as viewed, it isnot possible for the ray to pass on the right of all the edges of thepolygon, so if the w parameters, which are based on the 2D crossproducts, have the same sign for all edges of the polygon then thisindicates that the ray passes on the left of all the edges, such thatthe ray is determined to intersect the polygon. Hence it can be deducedthat, for a convex polygon of unknown orientation, if the w parametershave the same sign for all edges of the polygon then its orientation asviewed is indicated by any one of its edge signs.

Following step S318 the method passes to step S320 in which the polygontesting unit(s) 114 of the intersection testing module 108 outputs anindication of the result of the determination of whether the rayintersects the k^(th) polygon. This indication could be a binaryindication (e.g. a one-bit flag) to indicate either a ‘hit’ or a ‘miss’of the ray in respect of the polygon. In other examples, the indicationscould have different forms. Alongside an indication of whetherintersection occurs, there may be additional output attributes, e.g.,barycentric coordinates, orientation (clockwise/anticlockwise orfront/back facing), and/or an intersection distance.

In step S322 the polygon testing unit(s) 114 of the intersection testingmodule 108 determines whether there are more polygons to test. If so,then the method passes to step S324 in which the parameter, k, isincremented, and then the method passes back to step S310 so that stepsS310 to S320 can be performed for the next polygon in order to determinewhether the ray intersects the next polygon. When there are no morepolygons to test, the method passes to step S326. Step S326 can beperformed before, after or during step S322, i.e. step S326 can beperformed following step S320 irrespective of whether there are morepolygons to test.

In step S326, the outputted indication is used in the ray tracing system100 (e.g. by the processing logic 110) for rendering an image of a 3Dscene. For example, as well as the indication that the ray intersects apolygon, in step S320 the polygon intersection testing unit(s) 114 ofthe intersection testing module 108 can output an intersection distance(i.e. a distance between the ray origin and the point at which the rayintersects the polygon), and an indication of the position on thepolygon at which the ray intersects it (e.g. defined in barycentriccoordinates), and this information can be output to the processing logic110. In step S326 all the intersection tester output attributes,alongside other external attributes, e.g., scene data (e.g., lightsource(s), cube map), instance data, primitive/polygon ID, vertex data(e.g., positions, normal, texture coordinates), texture address(es),secondary ray data, etc., are processed to determine a colour valueand/or generate additional secondary rays for future traversal.

If barycentric coordinates are output in step S320, the magnitudes ofthe w parameters for the different edges of a polygon that the ray isdetermined to intersect may be used for determining the barycentriccoordinates. For example, where the polygon is a triangular polygonhaving vertices v₀, v₁ and v₂, for an intersection of a ray at anintersection point v with, the w values give twice the signed area ofthe sub-triangles formed by vertices (v, v₀, v₁), (v, v₁, v₂) and (v,v₂, v₀), up to uniform nonzero rescaling from the ray-space projection.The barycentric coordinates (which may be referred to as “arealcoordinates”) are found by normalising these three w weights (i.e.,summing them and dividing through by the total), so that they sum to 1.In particular, normalising the three w weights factors out all uniformscaling. As described above, all non-zero weights have the same sign foran intersection, so their magnitudes (rather than their signed values)can be normalised which reduces the complexity in the floating pointoperations, because it is simpler to add together three values which areknown to be nonnegative than to add three values whose signs areunknown. The resulting barycentric coordinates α, β and γ, correspondingto the edges (v₁, v₂), (v₂, v₀) and (v₀, v₁) respectively, define apoint of intersection of the ray with the convex polygont=O+αv₀+βv₁+γv₂. Finally, note that since the barycentrics, bydefinition, sum to one, only two of the three values need to be output(although, as described above, all three weights must be computed toperform the normalisation).

If intersection distances are output in step S320, these may becalculated in terms of ray lengths rather than actual Euclideandistances. For example, the length of the ray may be divided through by√{square root over (D_(x) ²+D_(y) ²+D_(z) ²)}, and the intersectiondistance can be determined in terms of ray-lengths from the ray origin.In other words, a value τ is computed, where t=O+Dτ is the point ofintersection of the ray with the convex polygon. This value may bedetermined using the barycentric coordinates according to theintersection equation Dτ=αv₀+βv₁+γv₂. In some examples, the projectedcoordinates s₀, s₁ and ₂ (where genericallys=A(D_(x)v_(x)+D_(y)v_(y)+D_(z)v_(z))) may be used to determine τ,according to the equation A(D_(x) ²+D_(y) ²+D_(z) ²)τ=αs₀+βs₁+γs₂(obtained by multiplying both sides of the intersection equation by theprojection matrix). The barycentric interpolated coordinates are thendivided by A(D_(x) ²+D_(y) ²+D_(z) ²) to obtain τ. For example, for arescaled ray with

$A = \frac{1}{D_{z}}$and for which values of

$\frac{D_{x}}{D_{z}},{\frac{D_{y}}{D_{z}}{and}\frac{1}{D_{z}}}$have been pre-computed and stored, the square rescaled ray length

$D^{\prime 2} = \left( {\left( \frac{D_{x}}{D_{z}} \right)^{2} + \left( \frac{D_{y}}{D_{z}} \right)^{2} + 1} \right)$may be calculated and then τ may be determined as

$\tau = {{\frac{1}{D_{z}} \cdot \frac{1}{D^{\prime 2}}}{\left( {{\alpha s_{0}} + {\beta s_{1}} + {\gamma s_{2}}} \right).}}$Here we see that the

$\frac{1}{D_{z}}$is required to prevent the intersection distance being overcounted by afactor of D_(z) and motivates the storage of this quantity in the rayformat. Note that this extra scaling factor is not required if we do notrescale the ray along the direction of the ray (with A=1) but may useadditional storage if we wish to retain the benefits of ray rescaling inthe remaining projection operations (e.g. so that we may encode both

$\frac{D_{x}}{D_{z}},\frac{D_{y}}{D_{z}}$as well as D_(x), D_(y) and D_(z)). Also note that in one example

${A = \frac{1}{\left( {D_{x}^{2} + D_{y}^{2} + D_{z}^{2}} \right)}},$which it precomputed avoids the need to rescale the barycentricinterpolated convex polygon coordinates but this too would requireadditional storage. In other examples, one component of the intersectionequation is used to determine τ, so that equivalentlyD_(x)τ=αv_(0x)+βv_(1x)+γv_(2x), D_(y)τ=αv_(0y)+βv_(1y)+γv_(2y) andD_(z)τ=αv_(0z)+βv_(1z)+γv_(2z). In examples where

$\frac{1}{D_{z}}{or}\frac{1}{❘D_{z}❘}$has been pre-computed and stored, it is often most efficient to selectthe last of these three options so that the intersection distance can becalculated by multiplying the barycentric interpolated vertex zcomponents by the pre-computed reciprocal. If a reciprocal basis isemployed (for which

$\frac{1}{D_{x}},\frac{1}{D_{y}}$as well as

$\frac{1}{D_{z}}$or some scaling thereof may be pre-computed and stored) then theprojection operations involve terms of

$\frac{v_{x}}{D_{x}},{\frac{v_{y}}{D_{y}}{and}{}\frac{v_{z}}{D_{z}}}$(which may be thought of as mapping the coordinates to “t” space) orpossibly, as in rescaled examples, terms involving

${D_{z}\frac{v_{x}}{D_{x}}},{D_{z}\frac{v_{y}}{D_{y}}}$and v_(z) (rescaled t space). Since we may write

$\begin{matrix}{\tau = {{\alpha\frac{v_{0x}}{D_{x}}} + {\beta\frac{v_{1x}}{D_{x}}} + {\gamma\frac{v_{2x}}{D_{x}}}}} \\{= {{\alpha\frac{v_{0y}}{D_{y}}} + {\beta\frac{v_{1y}}{D_{y}}} + {\gamma\frac{v_{2y}}{D_{y}}}}} \\{{= {{\alpha\frac{v_{0z}}{D_{z}}} + {\beta\frac{v_{1z}}{D_{z}}} + {\gamma\frac{v_{2z}}{D_{z}}}}},}\end{matrix}$it is possible to determine the intersection distance without anyfurther calculation beyond barycentric interpolation of the scaledvalues (that feature in the projection calculation) unless the ray hasbeen rescaled, in which case a factor of

$\frac{1}{D_{z}}$is needed as in the general non-reciprocal rescaled basis case.Alternatively, if for example it is not convenient to generate scaledvertex components (in t space) then the barycentric interpolated resultmay be scaled by the appropriate one of

$\frac{1}{D_{x}},{\frac{1}{D_{y}}{and}{}\frac{1}{D_{z}}}$(which may again involve a further scaling by

${\frac{1}{D_{z}}{if}\frac{D_{z}}{D_{x}}},{\frac{D_{z}}{D_{y}}{and}\frac{1}{D_{z}}}$have been pre-computed and stored). Whichever the case, both scaled andunscaled reciprocal bases provide the flexibility to select which axisshould be used for interpolation, which as mentioned earlier, allows anaxis to be chosen that has the most attractive error properties for agiven convex polygon (e.g. by selecting the axis with the minimum rangeof values across the convex polygon).

In the example shown in FIG. 3 , the data defining the ray and the datafor the vertices defining the polygon is read in step S302, and then instep S304 the x, y and z components of the ray and the vertices definingthe polygon are selectively permuted and/or reversed so thatD_(z)≥D_(x)≥0 and D_(z)≥D_(y)≥0. However, the ordering of parts of thesetwo steps (steps S302 and S304) may be different in different examples.For example, the selective permutation and/or reversing of the x, y andz components of the ray may be performed before the data is obtained. Itcan be useful to perform this selective permutation and/or reversing ofthe x, y and z components of the ray data before the data is read in, sothat we know that D_(z)≥D_(x)≥0 and D_(z)≥D_(y)≥0 before the data isread in. This allows some pre-computation of the ray data to beperformed and the pre-computed data can be stored in a store (e.g. amemory on the ray tracing unit 102) before it is read into theintersection testing module 108. This means that the processing involvedin this pre-computation can be performed once for a ray and usedmultiple times if the ray is involved in multiple intersection tests. Asdescribed above the pre-computed ray direction data that is read by theintersection testing module could comprise values of

$\frac{D_{x}}{D_{z}},\frac{D_{y}}{D_{z}}$and D_(z) or

$\frac{1}{D_{z}},$values of

$\frac{D_{z}}{D_{x}},\frac{D_{z}}{D_{y}}$and D_(z) or

$\frac{1}{D_{z}},$or values of

$\frac{1}{D_{x}},\frac{1}{D_{y}}$and D_(z) or

$\frac{1}{D_{z}}$to give just some examples, and in other examples, other values may bepre-computed and read in order to define the ray direction. Alongsidethe pre-computed ray data, an indication of the major axis (e.g., a2-bit enumeration), indicating the permutation, as well as the originalsigns of the ray direction components, indicating the reversal, isstored so that the same pre-computation can be applied to the polygonvertices when the ray is to be tested against them (and to post-modifythe orientation determination if appropriate). The (only) overhead hereis an extra 2 bits for indicating the major axis, but this (negligible)amount of additional storage is outweighed by avoiding finding the majorcomponent of the ray direction every time a polygon is tested againstthe ray.

As described above, an issue can arise in the intersection testingprocess if the ray intersects a point on an edge of a polygon. Forexample, if the ray intersects a point on a shared edge, or if the rayintersects a shared vertex of a closed fan, which is shared by multiplepolygons, then, in order for the intersection testing process to be“watertight”, the polygon intersection testing unit(s) 114 should, inmost cases, determine that the ray intersects at least one of thepolygons. In order for the intersection testing process to be“non-redundantly watertight”, the polygon intersection testing unit(s)114 should, in most cases, determine that the ray intersects one andonly one of the polygons. There are some exceptions to these definitionsof what is meant by “watertight” and “non redundantly watertight”intersection testing, as described in detail above. If the intersectiontesting process is not watertight then rendering errors may beintroduced into the rendered images, such as cracks in geometry, whichis not desirable. One way to achieve a “watertight” intersection test isto project onto a ray-dependent plane (e.g., given by ray-space basisvectors P and Q), followed by exact determination of whether thepolygon, including its boundary, cover the origin (e.g., by establishingwhether the origin is inside of, or strictly on, each edge). This is nota “non-redundantly watertight” intersection test as it results in allpolygons overlapping, such that all shared edges are hit two (or more)times and all shared vertices of closed fans are hit once for eachpolygon in the closed fan. The examples outlined below are configured toremove this redundancy.

In some examples described herein a plane vertex ordering scheme is usedto ensure that the intersection tests are “non-redundantly watertight”.The “plane vertex ordering scheme” may be referred to herein simply as a“vertex ordering scheme”. The plane vertex ordering scheme defines auniversal ordering of the 2D projected vertices which is independent ofthe winding order of the vertices (i.e., inherited from the ordered setsof vertices defining the convex polygons). The ordering of the projectedvertices according to the plane vertex ordering scheme is a total strictordering. This implies that, for two distinct projected vertices a andb, we either have a<b or a>b. Note that there is no “natural” totalstrict ordering of the plane (i.e., consistent with arithmeticoperations of the complex numbers) as there is for the line (i.e.,consistent with the arithmetic operations of the real numbers). Theordering of the projected vertices according to the plane vertexordering scheme defines a direction of each edge, independent of thedirection inherited from the winding order(s) of the polygon(s) whichinclude the edge. The winding order defines a cyclic ordering ofvertices, i.e., a binary relationship in and only in the followingcircumstances: v_(i)v_((i+1(mod n))) for n=0, . . . , n−1. Note thatthis is not a proper ordering as it does not meet all the axioms of apartial ordering, e.g., transitivity, and so is also not a totalordering like the projection-dependent ordering scheme, but is definedon the edges of the polygon and is used to determine their direction.Additionally, being a cyclic ordering, its binary relations arepreserved by rotations of the convex polygon, and are reversed byreflections of the convex polygon. In examples described herein, andduring intersection determination, an edge of a convex polygon may beconsidered part of the polygon's boundary precisely when the directionof the edge given by the plane vertex ordering scheme coincides with thedirection of the edge given by the winding order of the polygon.

In an example plane vertex ordering scheme, the universal direction of aprojected edge is from its vertex with smallest p component to itsvertex with largest p component, unless the projected vertices of theedge have equal p components, in which case the direction of a projectededge is from its vertex with smallest q component to its vertex withlargest q component. To state this more mathematically, a 2D edge hastwo vertices (v_(i)(with components p_(i) and q_(i)) and v_(j) (withcomponents p_(j) and q_(j))), and, under the plane vertex orderingscheme, the edge is directed from min(v_(i),v_(j)) to max(v_(i),v_(j)),wherein the vertices are ordered such that v_(i)<v_(j) if p_(i)<p_(j) orif (p_(i)=p_(j) and q_(i)<q_(j)), otherwise v_(i)>v_(j). To put thisanother way, a first projected vertex v_(i) with components, p_(i) andq_(i), along the respective axes of the pair of axes (P and Q), isbefore a second projected vertex v_(j) with components, p_(i) and q_(j),along the respective axes of the pair of axes, if p_(i)<p_(j) or if(p_(i)=p_(j) and q_(i)<q_(j)); whereas a first projected vertex v_(i)with components, p_(i) and q_(i), along the respective axes of the pairof axes, is after a second projected vertex v_(j) with components, p_(i)and q_(j), along the respective axes of the pair of axes, if p_(i)>p_(j)or if (p_(i)=p_(j) and q_(i)>q_(j)). Note that any nondegenerateprojected convex polygon (i.e., with a nonzero projected area) will haveat least three edges with nonequal endpoints. Hence, we either havev_(i)<v_(j) or v_(j)<v_(i) for all such edges, according to the planevertex ordering scheme, and hence a well-defined direction for all suchedges. In the specific example of a nondegenerate projected trianglepolygon we will have all three edges with nonequal endpoints, accordingthe plane vertex ordering scheme, and hence all three edges with awell-defined direction.

When two projected vertices do have the same p and q components thenthis means that either the edge is degenerate (i.e., equal componentsalong the axis parallel with the ray direction), or that the edge isnondegenerate (i.e. with different components along the axis parallelwith the ray direction vector) and the ray is looking along the edge ofthe convex polygon, i.e., its direction is perpendicular to the normalof the (planar) polygon (up to the accuracy of the projection). For aconvex polygon to be nondegenerate (in unprojected space and thereforealso in projected space), the former case must not occur for three ormore edges of the convex polygon. The latter case means the polygonappears as a line with respect to the ray, with zero projected area, andtherefore a convex polygon with such an edge is always degenerate (inprojected space). Such degenerate (projected) polygons may freely bemissed by the ray. It is acceptable for the rules of “watertight” or“non-redundantly watertight” intersection testers not to apply to suchdegenerate polygons, so long as they do apply when those polygons areomitted.

The plane vertex ordering scheme has eight alternative characterisationsgiving eight alternative pairs (for clockwise and anticlockwiseorientations) of edge tie-break behaviours. For convex polygons with aclockwise orientation the examples described herein exhibit a “Left-TopRule”, whereas for convex polygons with an anticlockwise orientation theexamples described herein exhibit a “Right-Bottom Rule”. Other examplesexhibit different tie-break behaviour pairs. For convex polygons of aclockwise/anticlockwise orientation, the eight alternative tie-breakbehaviour pairs exhibited are:v _(i) <v _(i) if p _(i) <p _(i) or if (p _(i) =p _(i) and q _(j) <q_(j)) (as described in the main examples herein)→“Left-Top/Right-BottomRule”  (i)v _(i) <v _(i) if p _(i) <p _(i) or if (p _(i) =p _(i) and q _(i) >q_(j))→“Right-Top/Left-Bottom Rule”  (ii)v _(i) <v _(i) if p _(i) >p _(i) or if (p _(i) =p _(i) and q _(j) <q_(j))→“Left-Bottom/Right-Top Rule”  (iii)v _(i) <v _(i) if p _(i) >p _(i) or if (p _(i) =p _(i) and q _(i) >q_(j))→“Right-Bottom/Left-Top Rule”  (iv)v _(i) <v _(i) if q _(j) <q _(i) or if (q _(j) =q _(j) and p _(i) <p_(j))→“Top-Left/Bottom-Right Rule”  (v)v _(i) <v _(i) if q _(i) >q _(j) OR if (q _(j) =q _(j) AND p _(i) <p_(j))→“Top-Right/Bottom-Left Rule”  (vi)v _(i)<v_(i) if q _(i) <q _(i) OR if (q _(j) =q _(j) AND p _(i) >p_(j))→“Bottom-Left/Top-Right Rule”  (vii)v _(i)<v_(i) if q _(i) >q _(j) OR if (q _(j) =q _(j) AND p _(i) >p_(j))→“Bottom-Right/Top-Left Rule”  (viii)

The above terms indicate which edges are considered part of the boundaryof a convex polygon in intersection determination. The first/secondtie-break behaviours correspond to when the convex polygon has anapparent clockwise/anticlockwise orientation viewed in projected 2Dspace. As the edge tie-break behaviour depends on thewinding/orientation of the convex polygon, we call it a“winding/orientation dependent” watertightness scheme. Note that thetie-break behaviour reverses for polygons having the oppositeorientation, e.g., “Left-Top Rule” and “Right-Bottom Rule”. For example,the term “Left-Top Rule” indicates that all edges bounding the convexpolygon from above, or bounding the convex polygon from the left butprecisely vertical, are considered part of the boundary of the polygon,and therefore are intersected. As the projection is ray dependent, theseedge tie-break behaviours apply to a single ray only, and may changefrom ray to ray, e.g., when there is a change of perceived orientationand/or major axis.

The way the edge tests are performed in examples described herein meansthat “non-redundant watertightness” can be considered to equate toanticommutativity on the result of the 2D cross product, with theproblem case being (signed) zero, corresponding to a polygon boundaryintersection. In other words, the watertightness problems can occur whena ray lands on an edge or a vertex (where f=±0). In this situation wewant to return only one hit. FIGS. 7 a and 7 b show a ray intersecting apoint 708 on an edge 706 that is shared between two polygons 702 and704. FIG. 7 a indicates the orientations of the two polygons from theray's perspective: both polygons 702 and 704 appear clockwise as viewed,which is inherited from their winding order. FIG. 7 a indicates thedirection of the edges of polygon 702, according to the ordering of thevertices in the ordered set of vertices defining the polygon 702, withclosed filled arrow heads. FIG. 7 a indicates the direction of the edgesof polygon 704, according to the ordering of the vertices in the orderedset of vertices defining the polygon 704, with open arrow heads. FIG. 7b indicates directions of the edges of the two polygons according to theplane vertex ordering scheme mentioned above, some of which differ fromthe directions of the edges given by the winding order in FIG. 7 a. Inthis example the polygons sharing the edge have consistent edge winding,i.e. with the two convex polygons 702 and 704 sharing the edge 706 withendpoints {v₁, v₂}, they are ordered (v₁, v₂) in one polygon (e.g.polygon 702) but are reversely ordered (v₂, v₁) in the contiguousneighbouring polygon (e.g. polygon 704), inherited from the respective(consistent) winding orders of the vertices in the ordered sets ofvertices defining the polygons. The examples described herein targetmanifold edges, i.e. situations in which two and only two polygons meetat a shared edge, permitting consistent winding of edges. Note that fora closed manifold mesh (i.e. all edges are shared and manifold)consistent edge winding is achievable only on orientable surfaces. For ashared silhouette edge, and assuming consistent edge winding, theperceived orientation of the polygons changes across the silhouette edge(but not the winding of the polygons), such that one of the polygons isfront-facing and the other of the polygons is back-facing. Implicitedges formed by the intersection of two polygons in their interiors,i.e. not at a shared edge or shared vertex on the polygons' boundaries(namely self-intersecting “non-simple” meshes), is excluded from theapplication of “non-redundant watertightness” and is not required toreturn a single hit only to achieve “non-redundant watertightness”.

In the examples shown in FIGS. 7 a and 7 b in which the ray isdetermined to intersect a point on an edge of a polygon then the wparameter is determined based upon whether the ordering, defined by thevertex ordering scheme (i.e. the “plane vertex ordering scheme”), of theprojected vertices defining the edge matches the ordering of thevertices in the ordered set of vertices defining the convex polygon(e.g. inherited from the winding order of the convex polygon). Inparticular, if which the ray is determined to intersect a point on anedge of a polygon then the sign of the w parameter for the edge of thepolygon is determined based upon whether the edge direction inheritedfrom the plane ordering of the projected vertices defining the edge,defined by the plane vertex ordering scheme, matches the edge directioninherited from the winding order of the vertices in the ordered set ofvertices defining the convex polygon. As mentioned above, the ray isdetermined to intersect a point on an edge if the 2D cross product ofthe projected positions of the two vertices defining the edge has amagnitude of (plus or minus) zero. If the ray intersects an edge (e.g.the edge 706 which is shared by polygons 702 and 704) then the planedirection of the edge is determined according to the plane vertexordering scheme, and this direction may be referred to as the “universalplane direction of the edge” (e.g. as seen in FIG. 7 b ). It is thendetermined whether the edge has a matching winding direction inheritedfrom the winding order of the polygon. If the polygons have consistentwinding orders (e.g. as seen by FIG. 7 a where both polygons 702 and 704have an apparent clockwise orientation as viewed) then the edge willhave a different winding direction for each of the two polygons: one ofthe two polygons sharing the edge will have a winding direction matchingthe universal plane direction of the edge, whereas the other polygonwill have a winding direction not matching the universal plane directionof the edge. The polygon intersection testing unit(s) 114 determinesthat the ray intersects with the polygon whose edge has a matchingwinding direction, and that the ray does not intersect with the polygonwhose edge does not have a matching winding direction.

For example, as shown in FIG. 7 b, the vertex v₁ is before the vertex v₂according to the plane vertex ordering scheme because p₁<p₂, so thedirection of the edge 706 according to the vertex ordering scheme isfrom v₁ to v₂. The winding direction of this edge in polygon 702 (whichappears as a clockwise orientation as viewed), inherited from thewinding ordering of the vertices v₁ and v₂ in the ordered set ofvertices (v₀, v₁, v₂) defining the polygon 702, matches the planeordering of the projected vertices v₁ and v₂ according to the planevertex ordering scheme. Therefore, the ray is determined to intersectpolygon 702 in the example shown in FIGS. 7 a and 7 b. In contrast, thewinding direction of this edge in polygon 704 (which also appears as aclockwise orientation as viewed), inherited from the winding order ofthe vertices v₂ and v_(i) in the ordered set of vertices (v₁, v₃, v₂)defining the polygon 704, does not match the plane ordering of theprojected vertices v₁ and v₂ according to the plane vertex orderingscheme. Therefore, the ray is determined to not intersect polygon 704 inthe example shown in FIGS. 7 a and 7 b. In summary, when the rayintersects the shared edge 706, it is determined to intersect one andonly one of the polygons, 702, and thus “non-redundant watertightness”is ensured in this example.

FIGS. 4 and 5 illustrate two examples for performing step S312 ofdetermining the w parameter to indicate which side of the e^(th) edge ofthe k^(th) polygon the ray passes on. In the first example (which may bereferred to as an “edge direction on-edge check” method), shown in FIG.4 , step S312 comprises step S402 in which the polygon intersectiontesting unit(s) 114 of the intersection testing module 108 determine a2D cross product f(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j), for the e^(th)edge which is defined by projected vertices v_(i) and v_(j), given intheir order in the ordered set of vertices defining the k^(th) polygon.

In step S404 the polygon intersection testing unit(s) 114 determineswhether f(v_(i),v_(j))=±0. If it is determined that f(v_(i),v_(j))≠±0,then this indicates that the ray does not intersect a point on thee^(th) edge, and then in step S406 the parameter, w, is set to equal tothe 2D cross product, i.e. w=f(v_(i),v_(j)). In this way, the sign ofthe w parameter for the e^(th) edge is determined to be the same as thesign of f(v_(i), v_(j)) if the ray is not determined to intersect apoint on the edge. Then the method passes to step S314.

However, if, in step S404, it is determined that f(v_(i),v_(j))=±0, thenthis indicates that the ray does intersect a point on the e^(th) edge,and then in step S408 the parameter, w, is set based on the planeordering of the projected vertices defining the edge according to theplane vertex ordering scheme. In particular, the parameter is determinedas w=(−1)^(v) ^(<f) ^(j) ·(+0). In other words, the magnitude of the wparameter is set to zero, and the sign of the w parameter is determinedto be the sign of (−1)^(v) ^(i) ^(<v) ^(j) , where v_(i)<v_(j)=1 if theprojected vertex v_(i) is before the projected vertex v_(j) according tothe plane vertex ordering scheme, and where v_(i)<v_(j)=0 if theprojected vertex v_(j) is before the projected vertex v_(i) according tothe plane vertex ordering scheme (note that either sign may be assignedfor a degenerate projected polygon where v_(i)=v_(j) since thewatertightness criterion need not apply). An alternative equivalentexpression for w is given by w=(−1)^(v) ^(j) ^(<v) ^(i) ·(−0).Alternative, but inequivalent, expressions for w are given by w=(−1)^(v)^(j) ^(<v) ^(i) ·(+0) and w=(−1)^(v) ^(i) ^(<v) ^(j) ·(−0), whichexhibit reverse behaviour. Then the method passes to step S314.

For the example method shown in FIG. 4 to be reliable, e.g. to guaranteeat least “watertight” behaviour for shared edges and shared vertices ofclosed fans, the particular implementation of the 2D cross productfunction f must be sign accurate when f(v_(i),v_(j))≠±0. In particular,the 2D cross product function f is anticommutative, so sgn (f(v_(i),v_(j)))=!sgn (f(v_(j), v_(i))), for all f(v_(i), v_(j))≠±0, and any signaccurate implementation must share this derived property. Furthermore,this property indicates that the sign of the implementation of f can becalculated only once for one polygon of a shared edge, and then thenegated result can be used for the other polygon of the shared edge(assuming consistent winding). It is reasonable to expect animplementation of f to be sign accurate for f≠±0, as it is equivalent tocorrectly determining whether p_(i)q_(j)<q_(i)p_(j) (negative) orp_(i)q_(j)>q_(i)p_(j) (positive) for all p_(i)q_(j)≠q_(i)p_(j). Also,for the “edge direction on-edge check” method, the kernel of theimplementation of the 2D cross product f must match the kernel of the 2Dcross product f, i.e., the sets of input values (restricted torepresentable floating points) mapped to ±0 by each function must match.This can be achieved by using an implementation of f with fully expandedintermediates and one instance of RAZ (round away from zero) rounding onthe result, or by using an extended range output. Other than at zero,the magnitude of w plays no part in guaranteeing “watertightness” or“non-redundant watertightness” for the example method shown in FIG. 4 .However, if the implementation of f shares the full anticommutativityproperty of f, i.e., that f(v_(i),v_(j))=−(f(v_(j),v_(i)), for allf(v_(i),v_(j))≠±0, then the result (both sign and magnitude) of theimplementation of f can be calculated only once for one polygon of ashared edge, and then the negated result can be used for the otherpolygon of the shared edge (assuming consistent winding).

It is noted that the IEEE single precision floating point formatallocates an encoding for both positive and negative zero (all zerosstarting with a zero or one respectively). An implementation of the 2Dcross product function f may generate −0 in place of +0, but this willbe implementation specific. In hardware, the precision of intermediatevalues may be expanded (be it implicitly or explicitly) to accommodate alarger range of values. It is clear that p_(i)q_(j)−q_(i)p_(j) cangenerate values outside the range of 8 bits of exponent and 23 bits ofmantissa, given four 32-bit floats as input. Given this, some roundingis performed if the result is to be output in 32 bits. If one of the twoclosest representable values to nonzero result w is 0, then w may alwaysbe rounded to +0 (generally considered to represent the traditionalunsigned 0) or may sometimes (or always) be rounded to −0. In otherexamples, the sign of w may be used to determine the rounding: If one ofthe two closest representative values to nonzero w is 0 and w ispositive, then w may usually be rounded to +0, and other times to −0. Ifone of the two closest representable value to nonzero w is 0 and w isnegative, then w may usually be rounded to −0, and other times to +0.If, in all such circumstances, the sign of nonzero w matches the sign of0 it is rounded to, then this is known as “sign accurate rounding”.Alternatively, where a nonzero w would be rounded to +0, it may insteadbe rounded to the minimal representable positive floating point value(e.g., when rounding away from zero), and where a nonzero w would berounded to −0, it may instead be rounded to the maximal representablenegative floating point value (e.g., when rounding away from zero). If wis exactly 0 then one would normally expect it to be encoded as +0, butmay sometimes (or always) be encoded as −0. The method described withreference to FIG. 4 , i.e., “edge direction on-edge check”, does notrequire the 2D cross product function f to be consistent in the signs ofexactly zero results, instead it overrides the sign in step S408.

In the example shown in FIGS. 7 a and 7 b, v₁<v₂ according to the planevertex ordering scheme. In polygon 702 the edge 706 is defined by theordered vertices v₁ and v₂, i.e. v₁=v₁ and v_(j)=v₂, so w is given anegative sign for the edge 706 in polygon 702. This is because in thiscase, w=(−1)^(v) ¹ ^(<v) ² . (+0)=(−1)¹·(+0)=−0. However, in polygon 704the edge 706 is defined by the ordered vertices v₂ and v₁, i.e. v₁=v₂and v_(j)=v₁, so w is given a positive sign for the edge 706 in polygon704. This is because in this case, w=(−1)^(v) ² ^(<v) ¹·(+0)=(−1)⁰·(+0)=+0.

In the second example (which may be referred to as a “full edge sorting”method), shown in FIG. 5 , step S312 comprises step S502 in which thepolygon intersection testing unit(s) 114 of the intersection testingmodule 108 determines an implementation of the 2D cross product withsorted input vertices f(max(v_(i), v_(j)),min(v_(i), v_(j))) for thee^(th) edge, defined by projected vertices v_(i) and v_(j) defining thek^(th) polygon. The two projected vertices, v_(i) and v_(j) defining theedge, are provided to the implementation of the 2D cross productfunction f in an order defined by the plane vertex ordering scheme. Inparticular, if the projected vertex v_(i) is before the projected vertexv_(j) (i.e. if v_(i)<v_(j)) according to the plane vertex orderingscheme then max(v_(i),v_(j))=v_(j) and min(v_(i),v_(j))=v_(i); whereasif the projected vertex v_(i) is after the projected vertex v_(j) (i.e.if v_(i)>v_(j)) according to the plane vertex ordering scheme thenmax(v_(i),v_(j))=v_(i) and min(v_(i),v_(j))=v_(j). By ordering inputvertices, by the plane ordering scheme, the 2D cross product f is onlydefined on half of the four-dimensional domain, possibly providing areasavings to any implementation taking advantage of this property.Furthermore, by sorting input vertices, it is evident that the result ofthis expression can be calculated only once per shared edge and then maybe duplicated across all polygons sharing that edge.

In step S504 the w parameter is determined as w=(−1)^(v) ^(i) ^(<v) ^(j)·f(max(v_(i),v_(j)),min(v_(i),v_(j))). More generally, the sign of thesigned parameter w for the edge is determined to be the same as the signof (−1)^(v) ^(i) ^(<v) ^(j) ·f(max(v_(i),v_(j)),min(v_(i),v_(j))), asthe magnitude of w plays no part in “watertightness” or “non-redundantlywatertightness” determination. An alternative expression for w, whichreverses the sign on the kernel of f but fixes the sign off the kernelof f, is given by w=(−1)^(v) ^(j) ^(<v) ^(i)·f(min(v_(i),v_(j)),max(v_(i),v_(j))). An alternative expression for w,which reverses the sign both on and off the kernel of f, is given byw=(−1)^(v) ^(j) ^(<v) ^(i) ·f(max(v_(i),v_(j)),min(v_(i),v_(j))). Analternative expression for w, which fixes the sign on the kernel of fbut reverses the sign off the kernel of f, is given by w=(−1)^(v) ^(i)^(<v) ^(j) ·f(min(v_(i),v_(j)),max(v_(i),v_(j))). Then the method passesto step S314.

A binary function f is anti-commutative if swapping the order of thevertices reverses the sign of f. The “full edge sorting” method shown inFIG. 5 lifts the restriction that the implementation of the 2D crossproduct function f be anti-commutative, whilst still ensuring that the wparameter is anti-commutative, and hence the result of this expressionmay be calculated only once for one polygon of a shared edge, and thenthe negated result may be used for the other polygon of the shared edge(assuming consistent winding). Also, the implementation of f need onlybe sign accurate on the domain v_(i)>v_(j) and, further still, need notmatch the kernel of f, lifting additional restrictions over the “Edgecheck on-edge method”. Although the implementation of f need not matchthe kernel of f, it must be consistent in sign, i.e. all positive or allnegative, for values that f maps to zero.

In the example shown in FIGS. 7 a and 7 b, v_(i)<v₂ according to theplane vertex ordering scheme, such that max(v₁,v₂)=v₂ and min(v₁,v₂)=v₁.Therefore, f(max(v₁,v₂),min(v₁,v₂))=f(v₂,v₁). In polygon 702 the edge706 is defined by the ordered vertices v₁ and v₂, i.e. v₁=v₁ andv_(j)=v₂, so w=(−1)¹·f(v₂,v₁)=−f(v₂,v₁) for the edge 706 in polygon 702.However, in polygon 704 the edge 706 is defined by the ordered verticesv₂ and v₁, i.e. v_(i)=v₂ and v_(j)=v₁, so w=(−1)⁰·f(v₂,v₁)=+f(v₂,v₁) forthe edge 706 in polygon 702. It can therefore be seen that the sign ofthe w parameter for edge 706 is different for polygons 702 and 704, suchthat the ray will be determined to intersect one (and only one) of thepolygons 702 and 704, namely 702, thereby ensuring “non redundantwatertightness” when the ray intersects the shared edge 706.

Ensuring “non-redundant watertightness” is a particular issue when atleast one vertex is a shared vertex which is used to define two or moreconvex polygons, e.g. in a closed fan. Methods described herein ensure“non-redundant watertightness” in the intersection testing when, for twoor more (e.g. exactly two) convex polygons, the ray is determined tointersect a point on an edge of one of those convex polygons: In theexample shown in FIGS. 7 a and 7 b the ray intersects a point somewherealong the interior of an edge shared by two polygons. In the exampleshown in FIGS. 8 a and 8 b the ray intersects a point on the end of anedge, in this case a vertex shared by multiple (e.g. five) polygonsforming a closed fan. In particular, FIGS. 8 a and 8 b show a rayintersecting a vertex v₂ that is shared between five polygons 802, 804,806, 808 and 810. FIG. 8 a indicates the winding orders of the fivepolygons, demonstrated by their clockwise orientation as viewed. FIG. 8a indicates the direction of the edges of polygon 802, according to theordering of the vertices in the ordered set of vertices defining thepolygon 802, with closed filled arrow heads. FIG. 8 a indicates thedirection of the edges of polygon 804, according to the ordering of thevertices in the ordered set of vertices defining the polygon 804, withopen arrow heads. FIG. 8 a indicates the direction of the edges ofpolygon 806, according to the ordering of the vertices in the orderedset of vertices defining the polygon 806, with closed unfilled arrowheads. FIG. 8 a indicates the direction of the edges of polygon 808,according to the ordering of the vertices in the ordered set of verticesdefining the polygon 808, with closed filled arrow heads. FIG. 8 aindicates the direction of the edges of polygon 810, according to theordering of the vertices in the ordered set of vertices defining thepolygon 810, with open arrow heads. FIG. 8 b indicates directions of theedges of the five polygons according to the plane vertex ordering schemementioned above. When determining which convex polygon is hit by a rayintersecting exactly with a shared vertex of a closed fan, the polygonintersection testing unit(s) 114 of the intersection testing module 108determines which of the polygons has two edges, both with the sharedvertex as an endpoint, whose plane directions (according to the planevertex ordering scheme) both match their winding directions (accordingto the winding order given by the ordered sets of vertices defining thepolygon). There will be one (and only one) such polygon provided thatthe polygons have consistent winding, form a (simple, manifold) closedfan, and are not part of a silhouette. Indeed, a vertex shared by such aclosed fan of (non-overlapping) convex polygons is bounded (in theplane) by the endpoints of the edges meeting at the shared vertex.Hence, given that the polygons have consistent winding, there will existone and only one polygon of the closed fan that has three orderedvertices according to its winding order, including the shared vertex asthe middle vertex, whose ordering matches the ordering of the planeordering scheme. In terms of the plane ordering exhibiting aLeft-Top/Right-Bottom tie-break behaviour pair, the polygon directlybelow the shared vertex (or below and to the right in the case of aprecisely vertical edge), e.g. 810, is intersected in the clockwisecase, and the polygon directly above the shared vertex (or above and tothe left in the case of a precisely vertical edge), e.g. 804, isintersected in the anticlockwise case. In FIGS. 8 a and 8 b, polygon 810has matching directions for its two edges which involve the sharedvertex, v₂. This is not the case for any of the polygons 802, 804, 806and 808. So, it is determined that the ray intersects with polygon 810and does not intersect with any of the polygons 802, 804, 806 or 808.

For example, in the example “edge direction on-edge check” methoddescribed above with reference to FIG. 4 , for the polygon 810 the wparameter for the edge defined by projected vertices v₀ and v₂ will bedetermined, in step S408, to be w=(−1)¹·(+0), which has a negative sign;and the w parameter for the edge defined by projected vertices v₂ and v₃will be determined, in step S408, to be w=(−1)¹·(+0), which has anegative sign. The w parameter for the edge defined by projectedvertices v₃ and v₀ will be determined, in step S406, to bew=f(v₃,v₀)=p₃q₀−q₃p₀, which will be a negative value (as can be seen byconsidering the winding direction of the edge v₃ and v₀, giving rise toa clockwise circulation relative to the origin, in FIG. 8 a ). Inaccordance with the vertex ordering scheme, since we have identified apair of intersected edges (zero magnitude w), both of which havenegative sign (i.e. such that the plane order matches the windingorder), it is determined that the ray intersects the polygon 810, andthat since the remaining nonzero w parameter is negative, the polygon810 has a clockwise orientation as viewed. In other examples, incontrast with the vertex ordering scheme, it may alternatively bedetermined that, since the w parameters for all the edges of polygon 810are negative, it is determined that the ray intersects the polygon 810,and hence also that it has a clockwise orientation as viewed.

It is noted that none of the other polygons 802 to 808 will have wparameters determined such that a pair of intersected edges both havenegative sign (and zero magnitude) or, as an immediate corollary, thatall edges have the same sign. For example, for polygon 802, the wparameter for the edge defined by projected vertices v₁ and v₂ will bedetermined, in step S408, to be w=(−1)¹·(+0), which has a negative sign;whereas the w parameter for the edge defined by projected vertices v₂and v₀ will be determined, in step S408, to be w=(−1)⁰·(+0), which has apositive sign, which is neither negative nor the same sign as the wparameter for the edge defined by the projected vertices v₁ and v². Forbrevity, we will not describe all of the calculations for polygons 804to 808 here because it should now be apparent that they will notdetermine w parameters with either both negative signs for eachintersected edge or having the same signs for all of their edges.Therefore the ray will not be determined to intersect with any ofpolygons 802, 804, 806 or 808.

It is noted that the vertex ordering scheme and the alternative checkfor sign consistency on all edges produce equivalent results forpolygons that have a clockwise orientation as viewed. That is to say,intersected clockwise polygons generate negative w parameters sopositive intersection results are both negative on intersected edges aswell as all remaining edges. Anticlockwise polygons, however, generatepositive w parameters, such that positive intersection results, in thecase of the vertex ordering scheme, retain negative sign on intersectededges, but positive sign on remaining edges, whereas the sign-equalitytest requires that both intersected edges and remaining edges all havepositive sign. With reference to FIG. 8 , if all triangles are specifiedin the reverse winding order such that they have anticlockwiseorientation as viewed, the sign of every w parameter is reversed. Thisimplies that, in polygon 804, the w parameter for the edge defined bythe projected vertices v₁ and v₂ will be determined, in step S408, to bew=(−1)¹·(+0), which has a negative sign, and that the w parameter forthe edge defined by the projected vertices v₂ and v₄ will also bedetermined to be w=(−1)¹·(+0). The w parameter for the edge defined bythe projected vertices v₄ and v₁ will be determined, in step S406 to bew=p₄q₁−q₄p₁, which will be a positive value (as the winding isanticlockwise). If the vertex ordering scheme is used, it will bedetermined that polygon 804 is the (only) intersected polygon (since allintersected edges have negative sign), whereas if sign-equality is used,it will be determined that polygon 804 is not intersected. As all wparameter signs are inverted, it will instead be determined that polygon810 is again intersected (as inversion doesn't affect equality),independent of orientation. We therefore refer to the sign-equality test(without further qualification) as a winding-independent scheme and thevertex ordering scheme as a winding-dependent scheme.

In the second example “full edge sorting” method described above withreference to FIG. 5 , for the polygon 802 the w parameter for the edgedefined by projected vertices v₁ and v₂ will be determined, in stepS504, to be w=−f(v₂,v₁)=∓α₂₁, the w parameter for the edge defined byprojected vertices v₂ and v₀ will be determined, in step S504, to bew=+f(v₂,v₀)=±α₂₀, and the w parameter for the edge defined by projectedvertices v₀ and v₁ will be determined, in step S504, to bew=−f(v₁,v₀)=−(+α₁₀)=−α₁₀. Similarly, for the polygon 804, the wparameter for the edge defined by projected vertices v₂ and v₁ will bedetermined, in step S504, to be w=+f(v₂,v₁)=±α₂₁, the w parameter forthe edge defined by projected vertices v₄ and v₂ will be determined, instep S504, to be w=+f(v₄,v₂)=±α₄₂, and the w parameter for the edgedefined by projected vertices v₁ and v₄ will be determined, in stepS504, to be w=−f(v₄,v₁)=−(+α₄₁)=−α₄₁. For the polygon 806 the wparameter for the edge defined by projected vertices v₂ and v₄ will bedetermined, in step S504, to be w=−f(v₄,v₂)=∓α₄₂, the w parameter forthe edge defined by projected vertices v₅ and v₂ will be determined, instep S504, to be w=+f(v₅,v₂)=+α₅₂, and the w parameter for the edgedefined by projected vertices v₄ and v₅ will be determined, in stepS504, to be w=−f(v₅,v₄)=−(+α₅₄)=−α₅₄=−(+a54)=−a54.). For the polygon 808the w parameter for the edge defined by projected vertices v₂ and v₅will be determined, in step S504, to be w=−f(v₅,v₂)=∓α₅₂, the wparameter for the edge defined by projected vertices v₃ and v₂ will bedetermined, in step S504, to be w=+f(v₃,v₂)=±α₃₂, and the w parameterfor the edge defined by projected vertices v₅ and v₃ will be determined,in step S504, to be w=+f(v₅,v₃)=+(−α₅₃)=−α₅₃. For the polygon 810 the wparameter for the edge defined by projected vertices v₂ and v₃ will bedetermined, in step S504, to be w=−f(v₃,v₂)=∓α₃₂, the w parameter forthe edge defined by projected vertices v₀ and v₂ will be determined, instep S504, to be w=−f(v₂,v₀)=∓α₂₀, and the w parameter for the edgedefined by projected vertices v₃ and v₀ will be determined, in stepS504, to be w=+f(v₃,v₀)=+(−α₃₀)=−α₃₀. In the above examples α₂₁, α₂₀,α₁₀, α₄₂, α₄₁, α₅₂, α₅₄, α₃₂, α₅₃ and α₃₀ are all nonnegative valuesand, as all pairs of inputs to the implementation of the 2D crossproduct are in the kernel of f, the sign indicators ± and ∓ areconsistent, i.e. they return either signs + and − respectively in allinstances, or signs − and + respectively in all instances. In this way,either polygon 810 is the one (and only one) polygon that is determinedto be intersected by the ray for respective signs + and −, or polygon804 is the one (and only one) polygon that is determined to beintersected by the ray for respective signs − and +, yielding threenegative values for w in either case (also indicating a clockwiseorientation as viewed).

Another example method for determining the w parameter for the e^(th)edge of the k^(th) polygon in step S312 (which may be referred to as a“endpoint sign/zero on-edge check” method) is described with referenceto the flow chart shown in FIG. 9 . Steps S902, S904 and S906 shown inFIG. 9 are the same as the steps S402, S404 and S406 shown in FIG. 4 .For the “endpoint sign/zero on-edge check”, polygons submitted to theintersection testing unit(s) 114 are assumed to be strictly convexpolygons.

Therefore, in step S902 the polygon intersection testing unit(s) 114 ofthe intersection testing module 108 determine an implementation of the2D cross product f(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j), for the e^(th)edge which is defined by projected vertices v_(i) and v_(j), orderedaccording to the winding order of the vertices in the ordered set ofvertices defining the k^(th) polygon. In step S904 the polygonintersection testing unit(s) 114 determine whether f(v_(i),v_(j))=±0. Ifit is determined that f(v_(i),v_(j))≠±0, then this indicates that theray should not intersect a point on the e^(th) edge, and then in stepS906 the parameter, w, is set to equal to the 2D cross product, i.e.w=f(v_(i),v_(j)). In this way, the sign of the w parameter for thee^(th) edge is determined to be the same as the sign of f(v_(i), v_(j))if it is determined that the ray should not intersect a point on theedge. Then the method passes to step S314. For the scheme described withreference to FIG. 9 , we make new assumptions on the implementation ofthe “2D cross product” f: (i) we assume that it is sign accurate off thekernel of f, e.g. if it shares the anticommutative property of the “2Dcross product” f, and (ii) we assume that we have one of the followingtwo types of implementation: any approximation where the kernel of theimplementation of the “2D cross product” equals the kernel of f (e.g. byrounding away from zero or using an extended range output), denoted a“round away from zero” (RAZ) type, or any approximation where the kernelof the implementation of the “2D cross product” is a strict superset ofthe kernel of f (e.g. where there is some instance of rounding underflowto zero), denoted a “round towards zero” (RTZ) type. In particular, itis assumed that if f outputs zero then any implementation approximatingit will also output zero, which is not a difficult requirement.

However, if, in step S904, it is determined that f(v_(i),v_(j))=±0, thenthis indicates that the ray might intersect a point on the e^(th) edge,and then in steps S908 to S912 the sign of the parameter, w, isdetermined using a module which comprises a lookup table (LUT) and/or alogic array. In examples described herein the module comprising thelookup table and/or logic array is implemented on the intersectiontesting module 108 as a block of pre-configured logic in fixed-functionhardware (e.g. as part of the polygon testing unit(s) 114). Thefunctionality of the lookup table and/or logic array described hereincan be considered to apply generally to the “module” comprising thelookup table and/or the logic array. The lookup table and/or logic arrayis configured to take, as its inputs, indications which classify each ofthe p_(i), q_(i), p_(j) and q_(j) coordinates as a trichotomy ofnegative, (plus or minus) zero or positive (e.g., as a “trit”), and tooutput, for valid combinations of classifications of the p_(i), q_(i),p_(j) and q_(j) coordinates, an indication of the sign of the signedparameter, w (e.g., as a bit). The lookup table or logic array may alsobe configured to take an indication of the sign of f and the booleanstate of an “underflow flag” as two other input bits.

The lookup table and/or logic array models a function with input tritsp_(i), q_(i), p_(j) and q_(j) (possibly as well as input bit sgn(f) andthe boolean state of an underflow flag) to provide an output bitindicating the sign of w. There are 2⁸¹ (or 2³²⁴) such functions. Thiscan be achieved in a first way by a look-up table, e.g. explicitlylisting 81 (or 324) bit values and using the inputs as a look-upindex/reference value. This would use at least 10 (or 40) bytes of rawstorage in a data store (e.g. hardcoded in software or a register inhardware). It is noted that, the lookup table is reduced to a quarter ofits size without the addition of the other two inputs. Alternatively,this can be achieved in a second way by a “logic array”, e.g. aconfiguration of logical operations in software or logical gates inhardware, performing the desired mapping. The symmetry of the problem,along with a small number of “don't care” outputs, allows the number oflogic gates to be reduced. Some hybrid of the two implementations(lookup table and logic array implementations) may also be possible,e.g. a reduced size LUT using some pre- or post-processing logic toleverage the symmetry of the problem. Either implementations may affordpower or area gains over the other, depending on the specificarchitecture, where the first way (LUT implementation) may be moresuitable to software, and the second (logic array implementation) may bemore suitable to hardware in general.

In step S908 the polygon testing unit(s) 114 classify each of the p_(i),q_(i), p_(j) and q_(j) coordinates for the vertices v_(i) and v_(j)defining the e^(th) edge as: (i) negative, (ii) (plus or minus) zero, or(iii) positive. It is noted that in order to be classified as negativeor positive, a coordinate is non-zero. In other words, for the purposesof this classification of the coordinates, (plus or minus) zero isconsidered to be non-negative and non-positive. Where the coordinatesare in a floating point format, and if denormals have been excluded orflushed, the exponent of a coordinate can be checked (e.g. all zeros) todetermine whether the coordinate is zero (regardless of the sign bit),and if the coordinate is not zero, the sign bit of the coordinate can bechecked to determine whether it is negative or positive. These are verysimple tests to perform, e.g. in fixed function hardware. If denormalnumbers are supported then the significand may be checked (e.g. allzeros), as well as the exponent, to determine whether a coordinate iszero.

In step S910 indications of the classifications of the coordinates areinput to the lookup table or logic array. For example, the indicationsof the classifications can be considered to be used as indices orreference values to lookup a piece of data (i.e. the sign of w) from thelookup table or to determine the sign of w using the logic array.

In step S912 the polygon testing unit(s) 114 receives the output fromthe lookup table or logic array which indicates the sign of the wparameter for the edge. The method then passes to step S314 andcontinues as described above.

The use of the lookup table or logic array makes the determination ofthe sign of the w parameter extremely fast to implement, so steps S908to S912 can be performed with very low latency and very low powerconsumption. A small amount of pre-configured logic can be used toimplement the lookup table or logic array so a small amount of siliconarea may be used to implement the lookup table or logic array inhardware, but this area will be very small because the operationsperformed by the lookup table or logic array are simple to implement inhardware.

Furthermore, the lookup table or logic array is configured such that ifthe order of the two projected vertices defining an edge is reversedthen the indication of the sign of the signed parameter that isoutputted from the lookup table or logic array is reversed. This ensuresthat the intersection testing is “non-redundantly watertight” fornon-silhouette shared edges of polygons with consistent winding orders.In examples described herein, this functionality is hardcoded into thelookup table or logic array.

As described above, the ray is determined to intersect a point on anedge if the implementation of the 2D cross productf(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j) of the positions of two projectedvertices defining the edge has a magnitude of zero.

For “endpoint sign/zero on-edge check”, whenf(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j)=±0, and in order to determine atie-break behaviour for “non-redundant watertightness”, the methodconsiders what the 2D cross product would be if the positions of the twoprojected vertices defining the edge were perturbed by ε=(ε_(p),ε_(q)),as illustrated in FIG. 10 . This perturbation indicates that theposition of the first of the projected vertices v_(i) defining the edgewould have coordinates (p_(i)+ε_(p)) and (q_(i)+ε_(q)) along therespective axes (P and Q) of the pair of axes of the coordinate systemof a ray, and that the position of the second of the projected verticesv_(j) defining the edge would have coordinates (p_(j)+ε_(p)) and(q_(j)+ε_(q)) along the respective axes (P and Q) of the pair of axes ofthe ray coordinate system. The perturbation being considered would betiny, but it would be big enough (i.e., by being considered nonzero),and at such a steep angle no representable edge could share, to move thevertices of an edge passing through the origin so that the edge would nolonger pass through the origin (such that the ray would no longerintersect the edge). Also, the perturbation being considered would betiny enough (i.e., by being considered smaller than any representablevector) not to move the vertices of an edge not passing through theorigin so that the edge would then pass through the origin (such thatthe ray would intersect the edge). Hence, with this consideration, itwould not be possible for a ray to exactly intersect an edge (includingits endpoints). Therefore, a tie-break behaviour is established for bothshared edges, where the interior of one (and only one) of the twopolygons sharing the edge would then cover the origin, and for sharedvertices, where, assuming a closed fan, the interior of one (and onlyone) of the multitude of polygons sharing the vertex would then coverthe origin. For the above assumptions to hold, the hypothetical valueε_(q) would be considered larger than zero but smaller than any positivereal value, and the hypothetical value ε_(p) would be considered largerthan zero but smaller than any positive rational expression involvingreal numbers and ε_(q).

The “endpoint sgn zero-check” method has eight alternativecharacterisations giving eight alternative edge tie-break behaviours.The examples described herein exhibit a “Right-Top Rule”. Other examplesexhibit different tie-break behaviour pairs. The eight alternativetie-break behaviour pairs exhibited are:0<ε_(p)<<ε_(q) (as described in detail in the main exampleherein)→“Right-Top Rule”  (i)ε_(p)<0<|ε_(p)|<<ε_(q)→“Left-Top Rule”  (ii)ε_(q)<0<ε_(p)<<|ε_(q)|→“Right-Bottom Rule”  (iii)ε_(q)<<ε_(p)<0→“Left-Bottom Rule”  (iv)0<ε_(q)<<ε_(p)→“Top-Right Rule”  (v)ε_(q)<0<ε_(p)→“Top-Left Rule”  (vi)ε_(p)<0<ε_(q)<<|ε_(p)|→“Bottom-Right Rule”  (vii)ε_(p)<<ε_(q)<0→“Bottom-Left Rule”  (viii)

The above terms indicate which edges are considered part of the boundaryof a convex polygon in intersection determination, regardless of theapparent orientation of the convex polygon viewed in projected 2D space.As the edge tie-break behaviour does not depend on thewinding/orientation of the convex polygon, we call it a“winding/orientation independent” watertightness scheme. For example,the term “Left-Top Rule” indicates that all edges bounding the convexpolygon from above, or bounding the convex polygon from the left butprecisely vertical, are considered part of the boundary of the polygon,and therefore are intersected. As the projection is ray dependent, theseedge tie-break behaviours apply to a single ray only, and may changefrom ray to ray, e.g., when there is a change of perceived orientationand/or major axis.

With this perturbation, the 2D cross product function becomes:f(v _(i) ,v _(j))=(p _(i)+ε_(p))(q _(j)+ε_(q))−(q _(i)+ε_(q))(p_(j)+ε_(p))=p _(i)q_(j) −q _(i) p _(j) +p _(i)ε_(q)+ε_(p) q _(j) −q_(i)ε_(p)−ε_(q) p _(j)  (1)

The lookup table is configured to output, for all valid combinations ofclassifications of the p_(i), q_(i), p_(j) and q_(j) coordinates, anindication of the sign of the signed parameter w. In some examples,e.g., fora RAZ implementation type, the lookup table or logic array isconfigured to output: (i) for some valid combinations of classificationsof the p_(i), q_(i), p_(j) and q_(j) coordinates, the sign of the signedparameter w as the sign, or opposite of the sign, of p_(i), q_(i), p_(j)or q_(j), and (ii) for some invalid combinations of classifications ofthe p_(i), q_(i), p_(j) and q_(j) coordinates, the sign of the signedparameter w as any value. In some other examples, e.g., for an RTZimplementation type, the lookup table or logic array is configured tofurther output: (iii) for some valid combinations of the classificationof the p_(i), q_(i), p_(j) and q_(j) coordinates, the sign of the signedparameter w as the sign, or opposite of the sign, of the product of twoof p_(i), q_(i), p_(j) or q_(j), and (iv) for one or more other validcombinations of classifications of the p_(i), q_(i), p_(j) and q_(j)coordinates, the sign of the signed parameter w, as the sign of theimplementation of the 2D cross product f. In addition, in some otherexamples, e.g., for an RTZ implementation type, an “underflow flag” isused to determine which of (i), (ii), (iii) or (iv) is configured to bethe output.

For example, the polygon intersection testing unit(s) 114 may operateaccording to a “round-away-from-zero” (RAZ) implementation type whendetermining the 2D cross product function for use in determining the wparameters, ensuring the kernel of the implementation of the 2D crossproduct matches the kernel of f. When a RAZ implementation type is usedthen the lookup table may be configured to output, for all validcombinations of classifications of the p_(i), q_(i), p_(j) and q_(j)coordinates for which f can be determined to be zero, an indication ofthe sign of the signed parameter, w. In particular, when a RAZ roundingmode is used then it is determined in step S904 thatf(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j)=±0 if and only if the value of the2D cross product truly is zero, and not because a non-zero value of the2D cross product has been approximated as zero by the implementation.Therefore, in this case, equation (1) becomes:f(v _(i) ,v _(j))=p _(i)ε_(q)+ε_(p) q _(j) −q _(i)ε_(p)−ε_(q) p_(j)  (2)

FIGS. 11 a to 11 l illustrate possible situations in which a rayintersects an edge of a strictly convex polygon in a RAZ type, and alsoillustrate what the lookup table or logic array is configured to outputas the sign of the w parameter for the different situations, as derivedfrom the form of equation (2).

FIGS. 11 a to 11 l illustrate the positions of the P and Q axes and(unperturbed) edges which intersect with the origin of the P and Q axes.Each edge is illustrated by a line connecting the vertices v_(i) tov_(j) which define the edge, with the arrowheads pointing in thedirection from v_(i) to v_(j), inherited from the winding order of astrictly convex polygon including that edge. The actual vertices v_(i)and v_(j) are not labelled in FIGS. 11 a to 11 l. FIGS. 11 a to 11 lalso show the exaggerated position of the origin 1102 relative to theedges if the edges were perturbed by ε=(ε_(p),ε_(q)), i.e. the origin isshifted by the opposite perturbation −ε=(−ε_(p),−ε_(q)) and it can beseen that if the perturbation was applied then the edges would no longerintersect the origin 1102.

In FIG. 11 a neither of the coordinates of the first vertex v_(i) arezero, i.e. p_(i)≠0 and q_(i)≠0, but the second projected vertex v_(j) isat the origin, i.e. p_(j)=q_(j)=0. FIG. 11 a shows four types of edgeswhich fall into this category for the four possible combinations ofsigns of p_(i), and q_(i). In this example, equation (2) becomesf(v_(i)v_(j))=p_(i)ε_(q)−q_(i)ε_(p), so since 0<ε_(p)<<ε_(q), the signof f will be the sign of p_(i). As such, the sign of the w parameter isset in this example to be equal to the sign of p_(i). The dual case,i.e. the reverse case, to the situation shown in FIG. 11 a is shown inFIG. 11 i, for which the sign of the w parameter is set to be theopposite of the sign of p_(j). The dual case demonstrates theanticommutative property of the tie-break behaviour as switching theorder of the input projected vertices, e.g. swapping every instance of iwith j in the subscripts of p and q, negates the sign of the result.

In FIG. 11 b the first vertex v_(i) is on the Q axis but not at theorigin, i.e. p_(i)=0 and q_(i)≠0, and the second projected vertex v_(j)is at the origin, i.e. p_(j)=q_(j)=0. FIG. 11 b shows two types of edgeswhich fall into this category, for the two possible signs of q_(i). Inthis example, equation (2) becomes f(v_(i),v_(j))=−q_(i)ε_(p), so thesign of f will be the opposite of the sign of q_(i). As such, the signof the w parameter is set in this example to be the opposite of the signof q_(i). The dual case, i.e. the reverse case, to the situation shownin FIG. 11 b is shown in FIG. 11 f, for which the sign of the wparameter is set to be the sign of q_(j).

In FIG. 11 c both vertices v_(i) and v_(j) are on the Q axis but not atthe origin, i.e. p_(i)=0, q_(i)≠0, p_(j)=0 and q_(j)≠0. Since, for astrictly convex polygon, a valid intersection can only occur when the(unperturbed) edge passes through the origin (i.e., an intersection witha strictly convex polygon cannot occur for an edge depicted in FIG. 11 k), then the edge must cross the P axis, and therefore q_(i) and q_(j)must have opposite sign in this example. FIG. 11 c shows two types ofedges which fall into this category, for the two valid combinations ofsigns of q_(i) and q_(j). In this example, equation (2) becomesf(v_(i),v_(j))=ε_(p)q_(j)−q_(i)lε_(p)=ε_(p)(q_(j)−q_(i)). Since q_(i)and q_(j) have opposite signs, the sign of (q_(j)−q_(j)) is equal to thesign of q_(j) and opposite to the sign of q_(i). Therefore the sign of fwill be the sign of q_(j) which is the opposite of the sign of q_(i). Assuch, the sign of the w parameter is set in this example to be the signof q_(j) or the opposite of the sign of q_(i). The situation shown inFIG. 11 c is its own dual case, i.e. its own reverse case, as the signof q_(j) is equal to the opposite of the sign of q_(i)

In FIG. 11 d the first vertex v_(i) is on the P axis but not at theorigin, i.e. p_(i)≠0 and q_(i)=0, and the second projected vertex v_(j)is at the origin, i.e. p_(j)=q_(j)=0. FIG. 11 d shows two types of edgeswhich fall into this category, for the two possible signs of p_(i). Inthis example, equation (2) becomes f(v_(i),v_(j))=p_(i)ε_(q), so thesign of f will be the sign of p_(i). As such, the sign of the wparameter is set in this example to be the sign of p_(i). The dual case,i.e. the reverse case, to the situation shown in FIG. 11 d is shown inFIG. 11 h, for which the sign of the w parameter is set to be theopposite of the sign of p_(j).

In FIG. 11 e neither of the coordinates of the first vertex v_(i) arezero, i.e. p_(i)≠0 and q_(i)≠0, and neither of the coordinates of thesecond vertex v_(j) are zero, i.e. p_(j)≠0 and q_(j)≠0. Since, for astrictly convex polygon, a valid intersection can only occur when the(unperturbed) edge passes through the origin (i.e., an intersection witha strictly convex polygon cannot occur for an edge depicted in FIG. 11 j), then the edge must cross the P axis so q_(i) and q_(j) must haveopposite sign in this example, and the edge must cross the Q axis sop_(i) and p_(j) must have opposite sign in this example too. FIG. 11 eshows four types of edges which fall into this category for the fourvalid combinations of signs of p_(i), q_(i), p_(j) and q_(j). In thisexample, equation (2) becomesf(v_(i),v_(j))=p_(i)ε_(q)+ε_(p)q_(j)−q_(i)ε_(p)−ε_(q)p_(j), so since0<ε_(p)<<ε_(q), the sign of f will be the sign of (p_(i)−p_(j)). Sincep_(i) and p_(j) have opposite signs, the sign of (p_(i)−p_(j)) is equalto the sign of p_(i) and opposite to the sign of p_(j). Therefore thesign of f will be the sign of p_(i) which is the opposite of the sign ofp_(j). As such, the sign of the w parameter is set in this example to bethe sign of p_(i) or the opposite of the sign of p_(j). The situationshown in FIG. 11 e is its own dual case, i.e. its own reverse case, asthe sign of p_(i) is equal to the opposite of the sign of p_(j).

In FIG. 11 f the first vertex v_(i) is at the origin, i.e.p_(i)=q_(i)=0, and the second projected vertex v_(j) is on the Q axisbut not at the origin, i.e. p_(j)=0 and q_(j)≠0. FIG. 11 f shows twotypes of edges which fall into this category, for the two possible signsof q_(j). In this example, equation (2) becomesf(v_(i),v_(j))=ε_(p)q_(j), so the sign of f will be the sign of q_(j).As such, the sign of the w parameter is set in this example to be thesign of q_(j). The dual case, i.e. the reverse case, to the situationshown in FIG. 11 f is shown in FIG. 11 b, for which the sign of the wparameter is set to be the opposite of the sign of q_(i).

In FIG. 11 g both vertices v_(i) and v_(j) are on the P axis but not atthe origin, i.e. p_(i)≠0, q_(i)=0, p_(j)≠0 and q_(j)=0. Since, for astrictly convex polygon, a valid intersection can only occur when the(unperturbed) edge passes through the origin (i.e., an intersection witha strictly convex polygon cannot occur for an edge depicted in FIG. 11 l), then the edge must cross the Q axis, and therefore p_(i) and p_(j)must have opposite sign in this example. FIG. 11 g shows two types ofedges which fall into this category, for the two valid combinations ofsigns of p_(i) and p_(j). In this example, equation (2) becomesf(v_(i),v_(j))=p_(i)ε_(q)−ε_(q)p_(j)=ε_(q)(p_(i)−p_(j)). Since p_(i) andp_(j) have opposite signs, the sign of (p_(i)−p_(j)) is equal to thesign of p_(i) and opposite to the sign of p_(j). Therefore the sign of fwill be the sign of p_(i) which is the opposite of the sign of p_(j). Assuch, the sign of the w parameter is set in this example to be the signof p_(i) or the opposite of the sign of p_(j). The situation shown inFIG. 11 g is its own dual case, i.e. its own reverse case, as the signof p_(i) is the opposite of the sign of p_(j).

In FIG. 11 h the first vertex v_(i) is at the origin, i.e.p_(i)=q_(i)=0, and the second projected vertex v_(j) is on the P axisbut not at the origin, i.e. p_(j)≠0 and q_(j)=0. FIG. 11 h shows twotypes of edges which fall into this category, for the two possible signsof p_(j). In this example, equation (2) becomesf(v_(i),v_(j))=−ε_(q)p_(j), so the sign of f will be the opposite of thesign of p_(j). As such, the sign of the w parameter is set in thisexample to be the opposite of the sign of p_(j). The dual case, i.e. thereverse case, to the situation shown in FIG. 11 h is shown in FIG. 11 d,for which the sign of the w parameter is set to be the sign of p_(i).

In FIG. 11 i the first projected vertex v_(i) is at the origin, i.e.p_(i)=q_(i)=0, but neither of the coordinates of the second vertex v_(j)are zero, i.e. p_(j)≠0 and q_(j)≠0. FIG. 11 i shows four types of edgeswhich fall into this category for the four possible combinations ofsigns of p_(j) and q_(j). In this example, equation (2) becomes f(v_(i),v_(j))=ε_(p)q_(j)−ε_(q)p_(j), so since 0<ε_(p)<<ε_(q), the sign of fwill be the opposite of the sign of p_(j). As such, the sign of the wparameter is set in this example to be opposite to the sign of p_(j).The dual case, i.e. the reverse case, to the situation shown in FIG. 11i is shown in FIG. 11 a, for which the sign of the w parameter is set tobe the sign of p_(i).

The situation shown in FIG. 11 j is potentially problematic as we areunable to ensure anticommutativity of the parameter w purely based onthe signs of p_(i), q_(i), p_(j) and q_(j), but as explained here it isactually not a problem. How anticommutativity of the parameter w cannotbe guaranteed is outlined: Similar to the situation shown in FIG. 11 e,neither of the coordinates of the first vertex v_(i) are zero, i.e.p_(i)≠0 and q_(i)≠0, and neither of the coordinates of the second vertexv_(j) are zero, i.e. p_(j)≠0 and q_(j)≠0. However, in the situationshown in FIG. 11 j, although the edge (if extended) would pass throughthe origin (such that p_(i)q_(j)−q_(i)p_(j)=0), the edge actually stopsbefore it reaches the origin. This situation is identified by seeingthat the p coordinates (and/or the q coordinates) for the two verticeshave the same sign, i.e. if p_(i) and p_(j) are both positive or areboth negative (indicated by the symbol ±_(p) in FIG. 11 j ), and ifq_(i) and q_(j) are both positive or are both negative (indicated by thesymbol ±_(q) in FIG. 11 j ), then we can be in the situation shown inFIG. 11 j. The potential problem with this situation is that the edgewinding can be reversed without changing any of the input signs, e.g. byrotating the edge, e.g. by 180 degrees. Hence, for this situation,winding cannot be purely a function of input signs of the p_(i), q_(i),p_(j) and q_(j) coordinates. However, this is not actually a problem as,for this situation, the determination of the sign of w for these edgesdoes not matter. This is because, where the polygon including this edgeis a strictly convex polygon, the polygon is either degenerate or theray does not intersect the strictly convex polygon no matter what signwe choose for this edge (e.g. when two of its edges return oppositesigns for their w parameters in the sign-inequality intersectioncriterion). As such, we can set the sign of the w parameter to beanything in this example and it will not cause any errors. However, asdescribed below, this situation is indistinguishable (purely based onthe signs of p_(i), q_(i), p_(j) and q_(j)) from the alternativesituation shown in FIG. 12 j, which may possibly occur when a RTZ typeimplementation is used to calculate f (and in this case rounds anunderflowing result to zero). As we have assumed that the implementationof the 2D cross product is sign accurate (e.g., by using “sign accuraterounding”) off of the kernel of f then, in the situation shown in FIG.12 j, the sign of f is correct. As such, the sign of the w parameter canbe set to be equal to the sign of f in both alternative situations 11 jand 12 j (when an RTZ type implementation is used to calculate f,otherwise any value may be used for w). The lookup table or logic arraymay receive the sign of f as an input, and set the sign of the wparameter to be equal to the sign of f. The situation shown in FIG. 11 jis its own dual case, i.e., its own reverse case.

The situation shown in FIG. 11 k is also potentially problematic, as weare also unable to ensure anticommutativity of the parameter w purelybased on the signs of p_(i), q_(i), p_(j) and q_(j), but as explainedhere it is actually not a problem. How anticommutativity of theparameter w cannot be guaranteed is outlined; similar to the situationin FIG. 11 c both vertices v_(i) and v_(j) are on the Q axis but not atthe origin, i.e. p_(i)=0, q_(i)≠0, p_(j)=0 and q_(j)≠0. However, in thesituation shown in FIG. 11 k, although the edge (if extended) would passthrough the origin (such that p_(i)q_(j)−q_(i)p_(j)=0) the edge actuallystops before it reaches the origin. This situation is identified byseeing that the q coordinates for the two vertices have the same sign,i.e. if q_(i) and q_(j) are both positive or are both negative, then weare in the situation shown in FIG. 11 k. The potential problem with thissituation is that the edge winding can be reversed without changing anyof the input signs, e.g. by rotating the edge, e.g. by 180 degrees.Hence, for this situation, winding cannot be purely a function of inputsigns of the p_(i), q_(i), p_(j) and q_(j) coordinates. However, this isnot actually a problem as, for this situation, the determination of thesign of w for these edges does not matter. This is because, where thepolygon including this edge is a strictly convex polygon, the polygon iseither degenerate or the ray does not intersect the strictly convexpolygon no matter what sign we choose for this edge (e.g. when two ofits edges return opposite signs for their w parameters in thesign-inequality intersection criterion). As such, we can set the sign ofthe w parameter to be anything in this example and it will not cause anyerrors (indicated by a X in FIG. 11 k ). The situation shown in FIG. 11k is its own dual case, i.e. its own reverse case.

The situation shown in FIG. 11 l is also potentially problematic, as weare also unable to ensure the anticommutativity of the parameter wpurely based on the signs of p_(i), q_(i), p_(j) and q_(j), but asexplained here it is actually not a problem. How anticommutativity ofthe parameter w cannot be guaranteed is outlined; similar to thesituation in FIG. 11 g both vertices v_(i) and v_(j) are on the P axisbut not at the origin, i.e. p_(i)≠0, q_(i)=0, p_(j)≠0 and q_(j)=0.However, in the situation shown in FIG. 11 l, although the edge (ifextended) would pass through the origin (such thatp_(i)q_(j)−q_(i)p_(j)=0) the edge actually stops before it reaches theorigin. This situation is identified by seeing that the p coordinatesfor the two vertices have the same sign, i.e. if p_(i) and p_(j) areboth positive or are both negative, then we are in the situation shownin FIG. 11 l. The potential problem with this situation is that the edgewinding can be reversed without changing any of the input signs, e.g. byrotating the edge, e.g. by 180 degrees. Hence, for this situation,winding cannot be purely a function of input signs of the p_(i), q_(i),p_(j) and q_(j) coordinates. However, this is not actually a problem as,for this situation the determination of the sign of w for these edgesdoes not matter. This is because, where the polygon including this edgeis a strictly convex polygon, the polygon is either degenerate or theray does not intersect the strictly convex polygon no matter what signwe choose for this edge (e.g. when two of its edges return oppositesigns for their w parameters in the sign-inequality intersectioncriterion). As such, we can set the sign of the w parameter to beanything in this example and it will not cause any errors (indicated bya X in FIG. 11 l ). The situation shown in FIG. 11 l is its own dualcase, i.e. its own reverse case.

FIGS. 11 a to 11 l illustrate all of the situations for combinations ofinputs to the lookup table or logic array for situations in which an(possibly extended) edge can exactly intersect the origin of the P and Qaxes. Therefore, when using an implementation of a RAZ type FIGS. 11 ato 11 l illustrate all of the situations in which the lookup table orlogic array is used, and indicate the outputs of the lookup table orlogic array for the different combinations of inputs.

In some other examples, the polygon intersection testing unit(s) 114 mayoperate according to an implementation of the 2D cross product that isnot a RAZ type for use in determining the w parameters. For example, theimplementation of the 2D cross product may use a rounding mode that isround to nearest, round towards zero, round towards positive infinity orround to negative infinity. When a non-RAZ implementation type is usedthen the value of f may be determined to be (plus or minus) zero eventhough the exact value of f is not zero. For example, a value of f thatis very close to zero may be rounded to (plus or minus) zero. This lossof precision in calculations involving floating point numbers isreferred to as “underflow”. Therefore, when a non-RAZ, but still signaccurate, implementation type (denoted RTZ) is used to calculate the 2Dcross product, then the lookup table or logic array needs to account forsituations in which the edge does not exactly intersect the origin (inaddition to the situations in which the edge does exactly intersect theorigin). One way to accomplish this is to provide an indication thatunderflow has occurred and in that case proceed to step S906 as if anonzero value had been computed or, equivalently, supply both anindication of underflow and the sign of f as additional inputs to thelookup table in step S910 such that the sign is used as output. Thereare, however, alternative methods that make uses of steps S908 and S910as now described. FIGS. 12 a to 12 j illustrate the positions of the Pand Q axes and edges which do not exactly intersect with the origin ofthe P and Q axes, but for which f may be determined to be zero due tothe kernel of its implementation being a strict superset of its ownkernel (e.g., due to underflow in the calculation). As with FIGS. 11 ato 11 l, each edge is illustrated in FIGS. 12 a to 12 j by a lineconnecting the vertices v_(i) to v_(j) which define the edge, with thearrowheads pointing in the direction from v_(i) to v_(j). The actualvertices v_(i) and v_(j) are not labelled in FIGS. 12 a to 12 j. FIGS.12 a to 12 j also show the position of the origin 1202, which does notappear perturbed, as shifting by ε_(p) and ε_(q) is negligible comparedto the nonzero values of p_(i), q_(i), p_(j) and q_(j) in theseunderflow cases.

In FIG. 12 a the first projected vertex v_(i) is on the Q axis but notat the origin, i.e. p_(i)=0 and q_(j)≠0, and the second projected vertexv_(j) is on the P axis but not at the origin, i.e. p_(j)≠0 and q_(j)=0.FIG. 12 a shows four types of edges which fall into this category forthe four possible combinations of signs of q_(i) and p_(j). In thisexample, the 2D cross product is f(v_(i),v_(j))=−q_(i)p_(j), so the signof f will be the opposite of the product of the signs of q_(i) andp_(j). As such, the sign of the w parameter is set in this example to bethe opposite of the product of the signs of q_(i) and p_(j). The dualcase, i.e. the reverse case, to the situation shown in FIG. 12 a isshown in FIG. 12 b, for which the sign of the w parameter is set to beequal to the product of the signs of p_(i) and q_(j).

When representing signs as + or −, indicating +1 or −1, we take theproduct of the signs to combine them and we negate to get the oppositeresult. When representing signs as 0 or 1, we take the XOR of the signs,or equivalently the sum of the signs modulo two, to combine them and weNOT, equivalently complement, to get the opposite result. The symbols inFIGS. 12 a to 12 i can indicate either of these respective operations,depending on the particular representation.

In FIG. 12 b the first projected vertex v_(i) is on the P axis but notat the origin, i.e. p_(i)≠0 and q_(i)=0, and the second projected vertexv_(j) is on the Q axis but not at the origin, i.e. p_(j)=0 and q_(j)≠0.FIG. 12 b shows four types of edges which fall into this category forthe four possible combinations of signs of p_(i) and q_(j). In thisexample, the 2D cross product is f(v_(i),v_(j))=p_(i)q_(j), so the signof f will be the equal to the product of the signs of p_(i) and q_(j).As such, the sign of the w parameter is set in this example to be equalto the product of the signs of p_(i) and q_(j). The dual case, i.e. thereverse case, to the situation shown in FIG. 12 b is shown in FIG. 12 a,for which the sign of the w parameter is set to be the opposite of theproduct of the signs of q_(i) and p_(j).

In FIG. 12 c the first projected vertex v_(i) is on the Q axis but notat the origin, i.e. p_(i)=0 and q_(i)≠0, and neither of the coordinatesof the second vertex v_(j) are zero, i.e. p_(j)≠0 and q_(j)≠0. FIG. 12 cshows eight types of edges which fall into this category for the eightpossible combinations of signs of q_(i), p_(j) and q_(j). In thisexample, the 2D cross product is f(v_(i),v_(j))=−q_(i)p_(j), so the signof f will be the opposite of the product of the signs of q_(i) andp_(j). As such, the sign of the w parameter is set in this example to bethe opposite of the product of the signs of q_(i) and p_(j). The dualcase, i.e. the reverse case, to the situation shown in FIG. 12 c isshown in FIG. 12 e, for which the sign of the w parameter is set to beequal to the product of the signs of p_(i) and q_(j).

In FIG. 12 d the first projected vertex v_(i) is on the P axis but notat the origin, i.e. p_(i)≠0 and q_(i)=0, and neither of the coordinatesof the second vertex v_(j) are zero, i.e. p_(j)≠0 and q_(j)≠0. FIG. 12 dshows eight types of edges which fall into this category for the eightpossible combinations of signs of p_(i), p_(j) and q_(j). In thisexample, the 2D cross product is f(v_(i),v_(j))=p_(i)q_(j), so the signof f will be the equal to the product of the signs of p_(i) and q_(j).As such, the sign of the w parameter is set in this example to be equalto the product of the signs of p_(i) and q_(j). The dual case, i.e. thereverse case, to the situation shown in FIG. 12 d is shown in FIG. 12 f,for which the sign of the w parameter is set to be the opposite of theproduct of the signs of q_(i) and p_(j).

In FIG. 12 e neither of the coordinates of the first vertex v_(i) arezero, i.e. p_(i)≠0 and q_(i)≠0, and the second projected vertex v_(j) ison the Q axis but not at the origin, i.e. p_(j)=0 and q_(j)≠0. FIG. 12 eshows eight types of edges which fall into this category for the eightpossible combinations of signs of p_(i), q_(i) and q_(j). In thisexample, the 2D cross product is f(v_(i),v_(j))=p_(i)q_(j), so the signof f will be the equal to the product of the signs of p_(i) and q_(j).As such, the sign of the w parameter is set in this example to be equalto the product of the signs of p_(i) and q_(j). The dual case, i.e. thereverse case, to the situation shown in FIG. 12 e is shown in FIG. 12 c,for which the sign of the w parameter is set to be the opposite of theproduct of the signs of q_(i) and p_(j).

In FIG. 12 f neither of the coordinates of the first vertex v_(i) arezero, i.e. p_(i)≠0 and q_(i)≠0, and the second projected vertex v_(j) ison the P axis but not at the origin, i.e. p_(j)≠0 and q_(j)=0. FIG. 12 fshows eight types of edges which fall into this category for the eightpossible combinations of signs of p_(i), q_(i) and p_(j). In thisexample, the 2D cross product is f(v_(i),v_(j))=−q_(i)p_(j), so the signof f will be the opposite of the product of the signs of q_(i) andp_(j). As such, the sign of the w parameter is set in this example to bethe opposite of the product of the signs of q_(i) and p_(j). The dualcase, i.e. the reverse case, to the situation shown in FIG. 12 f isshown in FIG. 12 d, for which the sign of the w parameter is set to beequal to the product of the signs of p_(i) and q_(j).

The situation shown in FIG. 12 g is potentially problematic, as we areunable to ensure the anticommutativity of the parameter w purely basedon the signs of p_(i), q_(i), p_(j) and q_(j), but as explained herethis problem can be overcome. How anticommutativity of the parameter wcannot be guaranteed is outlined; similar to the situation shown in FIG.11 e, neither of the coordinates of the first vertex v_(i) are zero,i.e. p_(i)≠0 and q_(i)≠0, and neither of the coordinates of the secondvertex v_(j) are zero, i.e. p_(j)≠0 and q_(j)≠0. In this case the signsof p_(i) and q_(i) (indicated by the symbols ±_(p) and ±_(q) in FIG. 12g ) match each other but are opposite to the signs of p_(j) and q_(j)(indicated by the symbols ∓_(p) and ∓_(q) in FIG. 12 g ). However, inthe situation shown in FIG. 12 g, the edge does not exactly pass throughthe origin, and instead f is determined to equal (plus or minus) zeroonly as the kernel of its implementation is a strict superset of its ownkernel (e.g. due to underflow in the calculation). The potential problemwith this situation is that the edge winding can be reversed withoutchanging any of the input signs, e.g. by passing the edge across theorigin. Hence, for this situation, winding cannot be purely a functionof input signs of the p_(i), q_(i), p_(j) and q_(j) coordinates. Inparticular, the signs of p_(i), q_(i), p_(j) and q_(j) cannotdistinguish the underflow case (shown in FIG. 12 g ) from a genuine f=0case (shown in FIG. 11 e ), so the situation shown in FIG. 12 g is nothandled correctly even with “sign accurate rounding”, as it will beoverridden by the sign determination of the situation in FIG. 11 e. Inorder to overcome this potential problem, the implementation of the 2Dcross product can flag up a case of underflow to detect this alternativesituation 12 g. This underflow flag can be passed as an additional inputbit to the lookup table or logic array, to choose between the signdetermination of the situation in FIG. 11 e, when the flag is not set,and the sign of the (sign accurate) determination of the 2D crossproduct for the situation in FIG. 12 g, when the flag is set. As analternative method, leveraging unused encodings in the floating-pointformat, when the 2D cross product identifies that f=0; that the signs ofthe inputs p_(i), q_(i), p_(j) and q_(j) coordinates are as shown inFIGS. 11 e and 12 g; and that underflow will occur (indicating thesituation in FIG. 12 g ); then, rather than rounding to (plus or minus)zero, the unrounded result of the 2D cross product can be encoded asspecial tokens −ε or +ε. Hence, in this alternative, and given that thef=0 on-edge case is avoided, the sign of the 2D cross product (i.e. thesign of either −ε or +ε) would be used as the sign of the w parameter,the same as for all f≠0 cases. The situation shown in FIG. 12 g is itsown dual case, i.e. its own reverse case, the same as the situation inFIG. 11 e.

The situation shown in FIG. 12 h is similar to the situation shown inFIG. 11 e as neither of the coordinates of the first vertex v_(i) arezero, i.e. p_(i)≠0 and q_(i)≠0, and neither of the coordinates of thesecond vertex v_(j) are zero, i.e. p_(j)≠0 and q_(j)≠0. However, itdiffers from the situation shown in FIG. 11 e as the signs of p_(i) andp_(j) are opposite to each other, and the signs of q_(i) and q_(j) matcheach other. In this example, the 2D cross product isf(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j), and since we know that the signsof p_(i) and p_(j) are opposite to each other, and the signs of q_(i)and q_(j) match each other, the sign of f will be equal to the productof the signs of p_(i) and q_(j), which is the opposite of the product ofthe signs q_(i) and p_(j). As such, the sign of the w parameter is setin this example to be equal to the product of the signs of p_(i) andq_(j), or the opposite of the product of the signs of q_(i) and p_(j).The situation shown in FIG. 12 h is its own dual case, i.e. its ownreverse case, as the sign of p_(i) is opposite to the sign of p_(j), andthe sign of q_(i) matches the sign of q_(j).

The situation shown in FIG. 12 i is similar to the situation shown inFIG. 11 e as neither of the coordinates of the first vertex v_(i) arezero, i.e. p_(i)≠0 and q_(i)≠0, and neither of the coordinates of thesecond vertex v_(j) are zero, i.e. p_(j)≠0 and q_(j)≠0. However, itdiffers from the situation shown in FIG. 11 e as the signs of p_(i) andp_(j) match each other, and the signs of q_(i) and q_(j) are opposite toeach other. In this example, the 2D cross product isf(v_(i),v_(j))=p_(i)q_(j)−q_(i)p_(j), and since we know that the signsof p_(i) and p_(j) match each other, and the signs of q_(i) and q_(j)are opposite to each other, the sign of f will be the opposite of theproduct of the signs of q_(i) and p_(j), which is the product of thesigns of p_(i) and q_(j). As such, the sign of the w parameter is set inthis example to be the opposite of the product of the signs of q_(i) andp_(j), or the product of the signs of p_(i) and q_(j). The situationshown in FIG. 12 i is its own dual case, i.e. its own reverse case, asthe sign of q_(i) is opposite to the sign of q_(j), and the sign ofp_(i) matches the sign of p_(j).

The situation shown in FIG. 12 j is potentially problematic, as we arealso unable to ensure the anticommutativity of the parameter w purelybased on the signs of p_(i), q_(i), p_(j) and q_(j), but as explainedhere this problem can be overcome. How anticommutativity of theparameter w cannot be guaranteed is outlined: Similar to the situationshown in FIG. 11 e, neither of the coordinates of the first vertex v_(i)are zero, i.e. p_(i)≠0 and q_(i)≠0, and neither of the coordinates ofthe second vertex v_(j) are zero, i.e. p_(j)≠0 and q_(j)≠0. In this casethe signs of p_(i) and p_(j) match each other, and (independently) thesigns of q_(i) and q_(j) match each other. However, in the situationshown in FIG. 12 g, the edge does not exactly pass through the origin,and instead the implementation of the 2D cross product f is determinedto equal (plus or minus) zero only as the kernel of its implementationis a strict superset of its own kernel (e.g. due to underflow in thecalculation). As with the situation shown in FIG. 11 j, the potentialproblem with this situation is that the edge winding can be reversedwithout changing any of the input signs, e.g. by rotating the edge, e.g.by 180 degrees. Hence, for this situation, winding cannot be purely afunction of input signs of the p_(i), q_(i), p_(j) and q_(j)coordinates. However, we can detect the situations shown in FIGS. 12 jand 11 j by checking that the implementation of the 2D cross product f=0and by checking that the input p_(i), q_(i), p_(j) and q_(j) coordinateshave appropriate signs. We have assumed that the implementation of the2D cross product function has “sign accurate rounding” off of the kernelof f meaning that, even in an underflow case, such as the situationshown in FIG. 12 j, the sign of the implementation of the 2D crossproduct matches the sign of f. As such, the sign of the w parameter canbe set to be equal to the sign of f. The lookup table or logic array mayreceive the sign of f as an input, and set the sign of the w parameterto be equal to the sign of f. Since the situations shown in FIGS. 11 jand 12 j are indistinguishable to the lookup table or logic array (i.e.the same input trits are received for the two cases), the sign of the wparameter is also set to be equal to the sign of f in the situationshown in FIG. 11 j (when an RTZ type implementation is used to calculatef, otherwise any value may be used for w) This is fine because, asdescribed above, the sign of w does not matter in the situation shown inFIG. 11 j, because an intersection cannot occur with a strictly convexpolygon including such an edge. The situation shown in FIG. 12 j is itsown dual case, i.e. its own reverse case.

There is a trichotomy (negative, (plus or minus) zero, positive) foreach of the four input coordinates. When the convex polygons aretriangles, the case of p_(i)=q_(i)=p_(j)=q_(j)=0 does not need to beincluded in the lookup table or logic array because, in general, anyprojected edge such that p_(i)=p_(j) and q_(i)=q_(j) represents adegenerate projected edge, i.e. it appears as a point, and trianglepolygons including such edges can be discarded because they must bedegenerate, and a ray does not need to be determined to intersectdegenerate polygons. So, in total, there are 3⁴−1=80 examples (excludingthe degenerate point line), and the FIGS. 11 a to 12 j illustrate whatsign for w is output from the lookup table or logic array for each ofthese classifications.

For a general convex polygon having n sides, at least n−2 projectededges must be degenerate for the projected polygon to be degenerate,i.e. to have zero projected area, and hence able to be culled freely. Ifthere are fewer than n−2 degenerate projected edges, then the signs ofdegenerate edge tests (i.e., the case of p_(i)=p_(j) and q_(i)=q_(j))can be ignored when testing whether all edge signs are equal, so thecase of p_(i)=q_(i)=p_(j)=q_(j)=0 does not need to be included in thelookup table of logic array for a general convex polygon either. Suchedges can be detected simply by checking if the two projected endpointsare equal. Geometrically, a higher order polygon with such degenerateedges collapses to a lower order polygon (similar to a non-strictlyconvex polygon). Such pathological cases can be avoided by ensuring thatno two vertices of a strictly convex polygon are equal. When the normalof the (planar) convex polygon is perpendicular to the ray direction itwill also have zero area as it appears as a line, and hence will also bea degenerate projected polygon, even if it has no degenerate (i.e.point) projected edges.

In one example, two bits can be used as the input for a coordinate toindicate negative, zero or positive. For example, a first bit couldindicate that the input coordinate is either zero or non-zero, and asecond bit could indicate that the input coordinate is either negativeor positive. In this example, in the situation that the first bitindicates that the input coordinate is zero, then it doesn't matter whatthe second bit is; whereas if the first bit indicates that the inputcoordinate is non-zero, then the second bit is used to indicate that theinput coordinate is either negative or positive. In another example, foreach of the four input coordinates, the three values of a trit (i.e., 0,1 and 2) can be assigned to encode the trichotomy of −, ±0 or + in anyorder. Then, the four trits (plus possibly an additional bit for thesign of the 2D cross product, plus possibly another additional bit forunderflow detection, both for an RTZ implementation type) can be mergedinto a value between 0 and 80 (e.g. representing a case ID between 1 and81 as shown in the table below) (or between 0 and 161, or 0 and 323), sothat the value can be encoded in binary with 7 (or 8, or 9) bits.

The lookup table or logic array may be hardcoded in fixed functioncircuitry in the polygon intersection testing unit(s) 114 of theintersection testing module 108. Table 2 below shows an example of theinputs and outputs of an implementation of the lookup table or logicarray, with some explanatory comments. A value of 0 for an inputcoordinate p_(i), q_(i), p_(j) or q_(j) indicates a value of either plusor minus zero. Likewise, a result of 0 for the implementation of the 2Dcross product f indicates a value of either plus or minus zero. For anyinput of output, a “don't care” value is represented by the symbol ‘X’:

TABLE 2 Example inputs, outputs and explanatory data for the lookuptable or logic array Inputs Dual Is Case Due to Case Underflow sgnOutput Case ƒ = ±0 or ID Flag Set? (ƒ) p_(i) q_(i) p_(j) q_(j) FigureSign of w ID Underflow? 1.0 X + − − − − 11j/12j + (RTZ type) 1.1 ƒ =0/Underflow X (RAZ type) 1.1 X − − − − − 11j/12j − (RTZ type) 1.0 ƒ =0/Underflow X (RAZ type) 2 X ± − − − 0 12f − 10 Underflow 3 X ± − − − +12i − 19 Underflow 4 X ± − − 0 − 12e + 28 Underflow 5 X ± − − 0 0 11a −37 ƒ = 0 6 X ± − − 0 + 12e − 46 Underflow 7 X ± − − + − 12h + 55Underflow 8 X ± − − + 0 12f + 64 Underflow 9.00 N + − − + + 11e − 73.01ƒ = 0 9.01 N − − − + + 11e − 73.00 ƒ = 0 9.10 Y + − − + + 12g + 73.11Underflow 9.11 Y − − − + + 12g − 73.10 Underflow 10 X ± − 0 − − 12d + 2Underflow 11 X ± − 0 − 0 11l X 11 ƒ = 0 12 X ± − 0 − + 12d − 20Underflow 13 X ± − 0 0 − 12b + 29 Underflow 14 X ± − 0 0 0 11d − 38 ƒ =0 15 X ± − 0 0 + 12b − 47 Underflow 16 X ± − 0 + − 12d + 56 Underflow 17X ± − 0 + 0 11g − 65 ƒ = 0 18 X ± − 0 + + 12d − 74 Underflow 19 X ± − +− − 12i + 3 Underflow 20 X ± − + − 0 12f + 12 Underflow 21.0 X + − + − +11j/12j + (RTZ type) 21.1 ƒ = 0/Underflow X (RAZ type) 21.1 X − − + − +11j/12j − (RTZ type) 21.0 ƒ = 0/Underflow X (RAZ type) 22 X ± − + 0 −12e + 30 Underflow 23 X ± − + 0 0 11a − 39 ƒ = 0 24 X ± − + 0 + 12e − 48Underflow 25.00 N + − + + − 11e − 57.01 ƒ = 0 25.01 N − − + + − 11e −57.00 ƒ = 0 25.10 Y + − + + − 12g + 57.11 Underflow 25.11 Y − − + + −12g − 57.10 Underflow 26 X ± − + + 0 12f − 66 Underflow 27 X ± − + + +12h − 75 Underflow 28 X ± 0 − − − 12c − 4 Underflow 29 X ± 0 − − 0 12a −13 Underflow 30 X ± 0 − − + 12c − 22 Underflow 31 X ± 0 − 0 − 11k X 31 ƒ= 0 32 X ± 0 − 0 0 11b + 40 ƒ = 0 33 X ± 0 − 0 + 11c + 49 ƒ = 0 34 X ± 0− + − 12c + 58 Underflow 35 X ± 0 − + 0 12a + 67 Underflow 36 X ± 0− + + 12c + 76 Underflow 37 X ± 0 0 − − 11i + 5 ƒ = 0 38 X ± 0 0 − 011h + 14 ƒ = 0 39 X ± 0 0 − + 11i + 23 ƒ = 0 40 X ± 0 0 0 − 11f − 32 ƒ =0 41 X ± 0 0 0 0 N/A X 41 ƒ = 0 42 X ± 0 0 0 + 11f + 50 ƒ = 0 43 X ± 00 + − 11i − 59 ƒ = 0 44 X ± 0 0 + 0 11h − 68 ƒ = 0 45 X ± 0 0 + + 11i −77 ƒ = 0 46 X ± 0 + − − 12c + 6 Underflow 47 X ± 0 + − 0 12a + 15Underflow 48 X ± 0 + − + 12c + 24 Underflow 49 X ± 0 + 0 − 11c − 33 ƒ =0 50 X ± 0 + 0 0 11b − 42 ƒ = 0 51 X ± 0 + 0 + 11k X 51 ƒ = 0 52 X ±0 + + − 12c − 60 Underflow 53 X ± 0 + + 0 12a − 69 Underflow 54 X ±0 + + + 12c − 78 Underflow 55 X ± + − − − 12h − 7 Underflow 56 X ± + − −0 12f − 16 Underflow 57.00 N + + − − + 11e + 25.01 ƒ = 0 57.01 N − + −− + 11e + 25.00 ƒ = 0 57.10 Y + + − − + 12g + 25.11 Underflow 57.11 Y− + − − + 12g − 25.10 Underflow 58 X ± + − 0 − 12e − 34 Underflow 59 X± + − 0 0 11a + 43 ƒ = 0 60 X ± + − 0 + 12e + 52 Underflow 61.0 X + +− + − 11j/12j + (RTZ type) 61.1 ƒ = 0/Underflow X (RAZ type) 61.1 X − +− + − 11j/12j − (RTZ type) 61.0 ƒ = 0/Underflow X (RAZ type) 62 X ± +− + 0 12f + 70 Underflow 63 X ± + − + + 12i + 79 Underflow 64 X ± + 0 −− 12d − 8 Underflow 65 X ± + 0 − 0 11g + 17 ƒ = 0 66 X ± + 0 − + 12d +26 Underflow 67 X ± + 0 0 − 12b − 35 Underflow 68 X ± + 0 0 0 11d + 44 ƒ= 0 69 X ± + 0 0 + 12b + 53 Underflow 70 X ± + 0 + − 12d − 62 Underflow71 X ± + 0 + 0 11l X 71 ƒ = 0 72 X ± + 0 + + 12d + 80 Underflow 73.00N + + + − − 11e + 9.01 ƒ = 0 73.01 N − + + − − 11e + 9.00 ƒ = 0 73.10Y + + + − − 12g + 9.11 Underflow 73.11 Y − + + − − 12g − 9.10 Underflow74 X ± + + − 0 12f + 18 Underflow 75 X ± + + − + 12h + 27 Underflow 76 X± + + 0 − 12e − 36 Underflow 77 X ± + + 0 0 11a + 45 ƒ = 0 78 X ± + +0 + 12e + 54 Underflow 79 X ± + + + − 12i − 63 Underflow 80 X ± + + + 012f − 72 Underflow 81.0 X + + + + + 11j/12j + (RTZ type) 81.1 ƒ =0/Underflow X (RAZ type) 81.1 X − + + + + 11j/12j − (RTZ type) 81.0 ƒ =0/Underflow X (RAZ type)

It can be seen that Table 2 has symmetry about its median value (i.e.,case 41), whereby an underflow case maintains its output sign (asrotating an edge by 180 degrees should fix the circulation of aninterior intersection), whereas an f=0 case reverses its sign (asrotating an edge by 180 degrees should reverse the circulation of anon-edge intersection), when going from case c to case 82−c. Table 2 alsohas symmetry in its duality: swapping the first two trits with thesecond two trits switches between a case and its dual case, reversingthe output sign, as required by the anticommutative property of thesigned area calculation. Both of these properties can be leveraged toreduce the size of a LUT implementing Table 2, by using a certain amountof pre- or post-processing logic, or to reduce the complexity of a logicarray implementing Table 1.

In the table above, the “dual case ID” column gives the case ID of thesituation in which the order of the vertices defining the edge has beenswapped. It can be seen that the dual case gives the opposite output forthe sign of the w parameter. This ensures that when a ray intersects ashared edge, it is determined to intersect one (and only one) of thepolygons which include the shared edge. In the final column, “f=0”indicates that this situation occurs because the exact value of the 2Dcross product is zero, i.e., no underflow has occurred (e.g. as shown inFIGS. 11 a to 11 l ). In the final column, “Underflow” indicates thatthis situation occurs due to underflow in the implementation of the 2Dcross product, even though the exact value of f is not zero (e.g. asshown in FIGS. 12 a to 12 j ). When a RAZ implementation type is usedthen the lookup table or logic array does not need to include the rowswhich can only occur due to underflow, and does need to include theunderflow flag or the sign of f as input columns. Hence, the lookuptable or logic array can be made smaller if the rounding mode is knownto be a RAZ implementation type, thereby reducing the amount of siliconneeded to hardcode the lookup table or logic array into the intersectiontesting module 108. In other words, in a RAZ implementation type thecases indicated as “Underflow” in the table above are not “valid”combinations of classifications of the p_(i), q_(i), p_(j) and q_(j)coordinates for the lookup table or logic array, i.e. they cannotvalidly occur when f=±0, as f and its implementation have matchingkernels. However, when an RTZ implementation type is used (and step S904hasn't been modified such that step S906 is performed if underflowoccurs) then the lookup table or logic array does include the rows whichcan only occur due to underflow (as well as all of the other rows), andthese cases do represent “valid” combinations of classifications of thep_(i), q_(i), p_(j) and q_(j) coordinates for the lookup table or logicarray, i.e. they can validly occur when f=±0, as the kernel of theimplementation of f is a strict superset of its own kernel. As describedin examples above, the underflow flag is set during the determination ofthe 2D cross product if the 2D cross product is determined to be notexactly equal to zero, but is rounded to zero. Entries in the lookuptable can be avoided for the RTZ examples by only invoking thewatertightness behaviours if both the magnitude of w is zero and theunderflow flag has not been set. In other words, in some examples, theray may be determined to intersect a point on an edge if: (i) the 2Dcross product of the positions of two projected vertices defining theedge has a magnitude of zero, and (ii) an underflow flag has not beenset during the determination of the 2D cross product. In these examples,if the 2D cross product of the positions of two projected verticesdefining the edge is determined to have a magnitude of zero but theunderflow flag has been set during the determination of the 2D crossproduct then the ray is not determined to intersect a point on an edge,and the sign of w for the edge may be set to be equal to the sign of the2D cross product (which is either +0 or −0).

For some of the rows of the lookup table or logic array the output valueis denoted by ‘X’ indicating a “don't care” result. In these examples,the sign of w does not matter because a strictly convex polygon withsuch an edge cannot be intersected by the ray, e.g. because the ray doesnot pass on the inside of one of the other edges of the polygon. Thecase in which p_(i)=q_(i)=p_(j)=q_(j)=0 is not required: either atriangle polygon including such an edge is degenerate, so can bediscarded, or such an edge of a higher order polygon can be identified,in order to exclude its sign from the intersection determination. Ineither case, it does not need to be included in the lookup table orlogic array.

In step S912 the output from the lookup table or logic array indicatingthe sign of the w parameter is received at the polygon intersectiontesting unit(s) 114 of the intersection testing module 108. Then themethod passes to step S314 which is described above, and the methodcontinues.

The method describing that an intersection is determined only if theuniversal plane direction matches the winding direction of a shared edgeensures “non-redundant watertightness” when the polygons forming a meshall have consistent winding orders, even on silhouette edges, and henceis denoted a “winding/orientation dependent” scheme, as mentionedearlier. The methods described with reference to FIGS. 4, 5 and 9 , whenused in conjunction with a sign-equality test in step S318, ensure“non-redundant watertightness” even when the polygons forming a mesh donot have consistent winding orders, but not on silhouette edges (wherezero or double hits may occur), and hence is denoted a“winding/orientation independent” scheme. In some implementations it issafe to assume that the geometry (i.e. the polygons) will be submittedto the ray tracing system 100 with consistent winding orders, e.g.orientable surfaces. However, it is possible that polygons may besubmitted to the ray tracing system 100 with inconsistent windingorders, e.g. non-orientable surfaces. In either situation “non-redundantwatertightness” cannot be ensured, using any method, universally (i.e.with arbitrary winding and both on and off silhouette edges), asdemonstrated by considering FIGS. 13 a to 13 d. However, by alternatingbetween a “winding/orientation dependent” and “winding/orientationindependent” scheme, we can select where zero and double hits occur,e.g. on or off the silhouette edge of the polygon mesh.

For example, the differentiating functionality between a“winding/orientation dependent” scheme and a “winding/orientationindependent” scheme is made clear in FIGS. 13 a and 13 b, which showthree polygon pairs (colour-coded black/grey), projected onto the PQplane, for both scheme types: FIG. 13 a corresponds to a“winding/orientation dependent” scheme; FIG. 13 b corresponds to a“winding/orientation independent” scheme. In FIGS. 13 a and 13 b a zerodenotes clockwise and a one denotes anticlockwise, for both theperceived orientation of a polygon and the circulation of a directededge. In each respective polygon pair, the directed edge going throughthe origin is considered clockwise when directed towards the right orprecisely up and considered anticlockwise when directed towards the leftor precisely down. This exhibits the behaviour of a “Left-Top Rule” forpolygons with a clockwise orientation.

The first (i.e. top left) respective polygon pairs in both FIGS. 13 aand 13 b denote the default configuration: a ray intersecting aconsistent shared edge (i.e., the edge has opposite winding direction inthe two polygons sharing the edge) for two polygons with clockwiseorientation. In this case the consistent shared edge is considered partof the bottom right polygon (the grey one) as all three edge booleansmatch for that polygon.

The second (i.e. top right) respective polygon pairs in both FIGS. 13 aand 13 b denote the configuration rotated by 180 degrees, with a raystill intersecting the same consistent shared edge. As a rotation hasbeen applied, the two polygons still have a clockwise orientation and,in either mode, the consistent shared edge is still considered part ofthe bottom right polygon (now the black one) as all three edge booleansstill match for that polygon.

The third (i.e. bottom left) respective polygon pairs in both FIGS. 13 aand 13 b denote the configuration reflecting along the consistent sharededge (e.g., when folding an edge so that it falls behind thesilhouette), with a ray still intersecting that same consistent sharededge. As a reflection has been applied, the two polygons now have ananticlockwise orientation. In the “winding/orientation independent”watertightness scheme (shown in FIG. 13 b ), the consistent edge isstill considered part of the bottom right polygon (now the black one) asall three edge booleans still match for that polygon (albeit withopposite value), so still exhibits the behaviour of a “Left-Top Rule”.In the “winding/orientation dependent” watertightness scheme (shown inFIG. 13 a ), the consistent shared edge is now considered part of thetop left polygon (now the grey one) as all three edge booleans now matchfor that polygon, so now exhibits the dual behaviour, i.e., that of a“Right-Bottom Rule”.

The difference in behaviour between the two types of scheme (shown inFIGS. 13 a and 13 b ) only comes into effect in the reflected case asthe orientation of the polygons has changed. One behaviour can beachieved from the other by XORing (an involution) the on-edge edgebooleans by the polygon orientation boolean (as described below).

FIG. 13 c shows a mesh of polygons 1302 to 1320 forming a circular band(i.e. annulus) with indications, by appropriate shading, of which edgesare considered to be part of each polygon in a “winding/orientationdependent” scheme (e.g. as described above with reference to the methodwhere an intersection is determined only if the universal planedirection matches the winding direction of a shared edge). FIG. 13 c isa view of the (unprojected) polygons from a viewpoint that is slightlyabove the camera's origin, for a camera firing primary rays away fromits origin. From the camera's origin only primitives 1302 to 1310,denoted the front polygons, may be visible, whereas primitives 1312 to1320, denoted the back polygons, may be occluded by the primitives infront of them. The silhouette boundary of the polygon mesh from theshown viewpoint is given by the boundary between the front and backpolygons, i.e. the shared edge between polygons 1302 and 1320 and theshared edge between polygons 1310 and 1312. A clockwise orientation of afront polygon has consistent winding with an anticlockwise orientationof a back polygon, and an anticlockwise orientation of a front polygon(e.g., polygon 1306) would have consistent winding with a clockwiseorientation of a back polygon (not shown in FIG. 13 c ). This can beappreciated by imagining rotating the band of polygons and consideringhow the orientation of each polygon changes as it transitions from beinga front polygon (i.e. in front of the silhouette) to a back polygon(i.e. behind the silhouette) and vice versa. The winding order of eachpolygon can be inferred from its perceived orientation in FIG. 13 c, andit can be seen that all of the polygons have a consistent winding order(i.e., each edge has opposite winding direction in the two polygonssharing the edge) except polygon 1306. The polygons are alternatelycolour coded grey or black for visual clarity only: the colour of apolygon is seen in FIG. 13 c by the colour of the winding symbol in theinterior of the polygon. When only a single hit occurs, a shared edgehas a solid line of either black or grey to indicate which of the blackpolygon or the grey polygon sharing the edge would be hit by a rayprecisely intersecting the shared edge. Note that the shared silhouetteedge between polygons 1302 and 1320, and also the shared silhouette edgebetween polygons 1310 and 1312, returns single hits only. When no hitoccurs, a dotted edge of a light grey colour indicates that no polygonshown in FIG. 13 c would be hit by a ray precisely intersecting theedge. When a double hit occurs, a shared edge has a dotted line of bothblack and grey to indicate that both the black polygon and the greypolygon sharing the edge would be hit by a ray precisely intersectingthe shared edge. For example, due to the inconsistent winding order ofpolygon 1306 relative to the other polygons, for the inconsistent sharededge between polygons 1304 and 1306, both of the polygons will bedetermined to be hit by a ray precisely intersecting the shared edge.Similarly, for the inconsistent shared edge between polygons 1306 and1308, both of the polygons will be determined to be hit by a rayprecisely intersecting the shared edge. In other examples, not shown inFIG. 13 c, zero hits may occur on shared edges in the interior (i.e. noton the silhouette boundary) of a polygon mesh due to inconsistentwinding. Double/zero hits can cause rendering redundancies/artefacts,respectively, at positions on such shared edges for a“winding/orientation dependent” scheme.

The occurrence of such rendering redundancies/artefacts described in theprevious example (which may occur due to inconsistent winding orders ofthe polygons forming a mesh) cannot be universally removed, but they canbe relocated so that they occur only at the silhouette edges of a meshof polygons. Visual artefacts at the silhouette edges of geometry in arendered image are less noticeable to a viewer of the rendered imagethan visual artefacts in the middle of an object which may, for examplein the instance of zero hits, give the appearance of the object having ahole in it, such that a background colour or occluded geometry can beseen through the object. Although visual artefacts are ameliorated,rendering redundancies, in the form or double hits, still occur.Therefore, overall, it can be preferable to move the possible renderingartefacts to the silhouette edges. This can be achieved in awinding/orientation independent scheme (such as the methods describedwith reference to FIGS. 4, 5 and 9 ), which is described with referenceto FIG. 13 d.

FIG. 13 d shows the same mesh of polygons as those shown in FIG. 13 c,which form a circular band (i.e. annulus) with indications, byappropriate shading, of which edges are considered to be part of eachpolygon in a “winding/orientation independent” scheme. To modify a“winding/orientation dependent” scheme to a “winding/orientationindependent” scheme, the same methods can be used for intersectiontesting, but with the addition of a post-processing XOR operation on thesign of the w parameter, by the sign of the w parameter of another edgeof the convex polygon, when the ray intersects a point on an edge of apolygon (i.e. when f=±0). For example, if the ray is determined tointersect a point on an edge of a convex polygon then, between stepsS314 and S318, there is an additional step performing an XOR operationusing: (i) the sign of the signed parameter, w, for the edge such thatf=±0 and (ii) the sign of the signed parameter, w, for an edge such thatf≠±0, indicating the perceived orientation of the convex polygon. Notethat, for any nondegenerate convex polygon (i.e., with nonzero projectedarea), there exists at least one edge such that f is not equal to zero(otherwise the projected area of the convex polygon would be zero, beinga sum of zeros). The sign of the signed parameter, w, being 0 could beused to indicate a clockwise orientation of a convex polygon, and thesign of the signed parameter, w, being 1 could be used to indicate ananticlockwise orientation of a convex polygon (or vice versa). Theresult of the XOR operation can be used as the sign bit of the wparameter, which can then be used in the determination of whether theray intersects the convex polygon (e.g., by checking that the signs ofall edges of a convex polygon are equal). By moving double/zero hits tothe silhouette boundary of a polygon mesh, this ensures “non-redundantwatertightness” within the interior of a mesh of polygons, even if thewinding orders of the polygons are inconsistent. However, it can resultin a lack of “non-redundant watertightness”, e.g., double/zero hits, ona silhouette edge of the mesh of polygons, regardless of the windingorders of the two polygons sharing the silhouette edge. For example, inFIG. 13 d there are no rendering artefacts within the interior (i.e. noton the silhouette boundary) of the mesh of polygons (e.g. no renderingartefacts on the edges of polygon 1306 even though it has aninconsistent winding order relative to the other polygons). However, onthe left silhouette shared edge (i.e. the edge shared by polygons 1302and 1320) both of the polygons 1302 and 1320 will be determined to behit by a ray precisely intersecting the shared edge, as indicated by theblack and grey shading of the dotted line. On the right silhouetteshared edge (i.e. the edge shared by polygons 1310 and 1312) neither ofthe polygons 1310 and 1312 will be determined to be hit by a rayprecisely intersecting the shared edge, as indicated by the light greyshading of the dotted line. Both behaviours occur even though polygons1302, 1310, 1312 and 1320 have consistent winding orders. However, insome applications, it is in fact desirable for a pair of intersectionsto occur through any convex object (e.g. in refraction) so that an entryand exit point are defined. In this case, the winding independent schemeguarantees the desired behaviour, whereas the winding dependent schemedoes not.

The post-processing XOR operation, being an involution, can be used bothto transform a “winding/orientation dependent” scheme into a“winding/orientation independent” scheme and to transform a“winding/orientation independent” scheme into a “winding/orientationdependent” scheme. Hence, this technique can be used to convert themethod, where an intersection occurs only if the universal planedirection matches the winding direction of a shared edge, into a“winding/orientation independent” scheme, and also used to convert themethods as described above with reference to FIGS. 4, 5 and 9 into a“winding/orientation dependent scheme”.

In some examples of a “winding/orientation independent” scheme (e.g.,the methods as described above with reference to FIGS. 4, 5 and 9 ), aray is determined to intersect a convex polygon if and only if the signof the w parameter is equal for all edges of the convex polygon. E.g.,for a triangle polygon, we have an intersection if and only ifsgn(w₀)=sgn(w₁)=sgn(w₂) for signed parameters w₀, w₁ and w₂,corresponding to the three edges of the triangle. After modifying thescheme to be “winding dependent”, if w₀ is the signed parameter of anon-edge case (i.e., f=±0) and w₁ is the signed parameter of an off-edgecase (i.e., f≠±0) then, with the post-processing XOR operation, theintersection determination becomes sgn(w₀) ⊕ sgn(w₁)=sgn(w₁)=sgn(w₂).This is satisfied only if sgn(w₀)=0. In other words, for an intersectionto occur, the sign of the signed parameter w₀ for the on-edge case mustbe 0, regardless of the sign of the signed parameter of any other edge.In this scenario, the intersection logic can be re-expressed as(sgn(w₀)=0) ^ (sgn(w₁)=sgn(w₂)). If instead, for an intersection tooccur, the sign of the signed parameter w₀ must be 1 then the “windingdependent” scheme exhibits the reverse tie-breaking behaviour foron-edge cases. Note that this is consistent with the intersectioncriterion described earlier that treats edge intersections as specialcases, in place of the simple sign-equality test. Here we have describeda method to reconcile the two approaches with a flexible over-archingscheme.

By augmenting the on-edge signs by the sign of the polygon (denoting itsperceived orientation) in the “winding/orientation dependent” scheme,the tie-breaking behaviour reverses between a polygon with a sign of 0and a polygon with a sign of 1. The eight characterisations of thetie-breaking behaviour split into four pairs: “Top-Left Rule” and“Bottom-Right Rule”, “Top-Right Rule” and “Bottom-Left Rule”, “Left-TopRule” and “Right-Bottom Rule”, and “Right-Top Rule” and “Left-BottomRule”. One member of each of these tie-break behaviour pairs correspondsto a polygon with a sign of 0, and the other to a polygon with a signof 1. This is demonstrated in FIG. 13 c where a “Right-Bottom Rule”applies to polygon 1306, whereas a “Left-Top Rule” applies to all otherpolygons.

In summary, if consistent winding orders of polygons are expected (oreven guaranteed) to be submitted, e.g., orientable surfaces, e.g., anannulus, then the “winding/orientation dependent” scheme may bepreferable, whereas if inconsistent winding orders of polygons areexpected (or even impossible to avoid) to be submitted, e.g.,non-orientable surfaces, e.g., a Mobius band, then the“winding/orientation independent” scheme may be preferable.

FIG. 14 shows a computer system in which the ray tracing systemsdescribed herein may be implemented. The computer system comprises a CPU1402, a GPU 1404, a memory 1406 and other devices 1414, such as adisplay 1416, speakers 1418 and a camera 1422. A ray tracing unit 1410(corresponding to ray tracing unit 102) is implemented on the GPU 1404,as well as a Neural Network Accelerator (NNA) 1411. In other examples,the ray tracing unit 1410 may be implemented on the CPU 1402 or withinthe NNA 1411 or as a separate processing unit in the computer system.The components of the computer system can communicate with each othervia a communications bus 1420. A store 1412 (corresponding to memory104) is implemented as part of the memory 1406.

The ray tracing system of FIG. 1 is shown as comprising a number offunctional blocks. This is schematic only and is not intended to definea strict division between different logic elements of such entities.Each functional block may be provided in any suitable manner. It is tobe understood that intermediate values described herein as being formedby a ray tracing system need not be physically generated by the raytracing system at any point and may merely represent logical valueswhich conveniently describe the processing performed by the ray tracingsystem between its input and output.

The ray tracing units, and specifically the intersection testing modulesdescribed herein may be embodied in hardware on an integrated circuit.The intersection testing modules described herein may be configured toperform any of the methods described herein. Generally, any of thefunctions, methods, techniques or components described above can beimplemented in software, firmware, hardware (e.g., fixed logiccircuitry), or any combination thereof. The terms “module,”“functionality,” “component”, “element”, “unit”, “block” and “logic” maybe used herein to generally represent software, firmware, hardware, orany combination thereof. In the case of a software implementation, themodule, functionality, component, element, unit, block or logicrepresents program code that performs the specified tasks when executedon a processor. The algorithms and methods described herein could beperformed by one or more processors executing code that causes theprocessor(s) to perform the algorithms/methods. Examples of acomputer-readable storage medium include a random-access memory (RAM),read-only memory (ROM), an optical disc, flash memory, hard disk memory,and other memory devices that may use magnetic, optical, and othertechniques to store instructions or other data and that can be accessedby a machine.

The terms computer program code and computer readable instructions asused herein refer to any kind of executable code for processors,including code expressed in a machine language, an interpreted languageor a scripting language. Executable code includes binary code, machinecode, bytecode, code defining an integrated circuit (such as a hardwaredescription language or netlist), and code expressed in a programminglanguage code such as C, Java or OpenCL. Executable code may be, forexample, any kind of software, firmware, script, module or librarywhich, when suitably executed, processed, interpreted, compiled,executed at a virtual machine or other software environment, cause aprocessor of the computer system at which the executable code issupported to perform the tasks specified by the code.

A processor, computer, or computer system may be any kind of device,machine or dedicated circuit, or collection or portion thereof, withprocessing capability such that it can execute instructions. A processormay be or comprise any kind of general purpose or dedicated processor,such as a CPU, GPU, NNA, System-on-chip, state machine, media processor,an application-specific integrated circuit (ASIC), a programmable logicarray, a field-programmable gate array (FPGA), or the like. A computeror computer system may comprise one or more processors.

It is also intended to encompass software which defines a configurationof hardware as described herein, such as HDL (hardware descriptionlanguage) software, as is used for designing integrated circuits, or forconfiguring programmable chips, to carry out desired functions. That is,there may be provided a computer readable storage medium having encodedthereon computer readable program code in the form of an integratedcircuit definition dataset that when processed (i.e. run) in anintegrated circuit manufacturing system configures the system tomanufacture an intersection testing module configured to perform any ofthe methods described herein, or to manufacture an intersection testingmodule comprising any apparatus described herein. An integrated circuitdefinition dataset may be, for example, an integrated circuitdescription.

Therefore, there may be provided a method of manufacturing, at anintegrated circuit manufacturing system, an intersection testing moduleas described herein. Furthermore, there may be provided an integratedcircuit definition dataset that, when processed in an integrated circuitmanufacturing system, causes the method of manufacturing an intersectiontesting module to be performed.

An integrated circuit definition dataset may be in the form of computercode, for example as a netlist, code for configuring a programmablechip, as a hardware description language defining hardware suitable formanufacture in an integrated circuit at any level, including as registertransfer level (RTL) code, as high-level circuit representations such asVerilog or VHDL, and as low-level circuit representations such as OASIS®and GDSII. Higher level representations which logically define hardwaresuitable for manufacture in an integrated circuit (such as RTL) may beprocessed at a computer system configured for generating a manufacturingdefinition of an integrated circuit in the context of a softwareenvironment comprising definitions of circuit elements and rules forcombining those elements in order to generate the manufacturingdefinition of an integrated circuit so defined by the representation. Asis typically the case with software executing at a computer system so asto define a machine, one or more intermediate user steps (e.g. providingcommands, variables etc.) may be required in order for a computer systemconfigured for generating a manufacturing definition of an integratedcircuit to execute code defining an integrated circuit so as to generatethe manufacturing definition of that integrated circuit.

An example of processing an integrated circuit definition dataset at anintegrated circuit manufacturing system so as to configure the system tomanufacture an intersection testing module will now be described withrespect to FIG. 15 .

FIG. 15 shows an example of an integrated circuit (IC) manufacturingsystem 1502 which is configured to manufacture an intersection testingmodule as described in any of the examples herein. In particular, the ICmanufacturing system 1502 comprises a layout processing system 1504 andan integrated circuit generation system 1506. The IC manufacturingsystem 1502 is configured to receive an IC definition dataset (e.g.defining an intersection testing module as described in any of theexamples herein), process the IC definition dataset, and generate an ICaccording to the IC definition dataset (e.g. which embodies anintersection testing module as described in any of the examples herein).The processing of the IC definition dataset configures the ICmanufacturing system 1502 to manufacture an integrated circuit embodyingan intersection testing module as described in any of the examplesherein.

The layout processing system 1504 is configured to receive and processthe IC definition dataset to determine a circuit layout. Methods ofdetermining a circuit layout from an IC definition dataset are known inthe art, and for example may involve synthesising RTL code to determinea gate level representation of a circuit to be generated, e.g. in termsof logical components (e.g. NAND, NOR, AND, OR, MUX and FLIP-FLOPcomponents). A circuit layout can be determined from the gate levelrepresentation of the circuit by determining positional information forthe logical components. This may be done automatically or with userinvolvement in order to optimise the circuit layout. When the layoutprocessing system 1504 has determined the circuit layout it may output acircuit layout definition to the IC generation system 1506. A circuitlayout definition may be, for example, a circuit layout description.

The IC generation system 1506 generates an IC according to the circuitlayout definition, as is known in the art. For example, the ICgeneration system 1506 may implement a semiconductor device fabricationprocess to generate the IC, which may involve a multiple-step sequenceof photo lithographic and chemical processing steps during whichelectronic circuits are gradually created on a wafer made ofsemiconducting material. The circuit layout definition may be in theform of a mask which can be used in a lithographic process forgenerating an IC according to the circuit definition. Alternatively, thecircuit layout definition provided to the IC generation system 1506 maybe in the form of computer-readable code which the IC generation system1506 can use to form a suitable mask for use in generating an IC.

The different processes performed by the IC manufacturing system 1502may be implemented all in one location, e.g. by one party.Alternatively, the IC manufacturing system 1502 may be a distributedsystem such that some of the processes may be performed at differentlocations, and may be performed by different parties. For example, someof the stages of: (i) synthesising RTL code representing the ICdefinition dataset to form a gate level representation of a circuit tobe generated, (ii) generating a circuit layout based on the gate levelrepresentation, (iii) forming a mask in accordance with the circuitlayout, and (iv) fabricating an integrated circuit using the mask, maybe performed in different locations and/or by different parties.

In other examples, processing of the integrated circuit definitiondataset at an integrated circuit manufacturing system may configure thesystem to manufacture an intersection testing module without the ICdefinition dataset being processed so as to determine a circuit layout.For instance, an integrated circuit definition dataset may define theconfiguration of a reconfigurable processor, such as an FPGA, and theprocessing of that dataset may configure an IC manufacturing system togenerate a reconfigurable processor having that defined configuration(e.g. by loading configuration data to the FPGA).

In some embodiments, an integrated circuit manufacturing definitiondataset, when processed in an integrated circuit manufacturing system,may cause an integrated circuit manufacturing system to generate adevice as described herein. For example, the configuration of anintegrated circuit manufacturing system in the manner described abovewith respect to FIG. 15 by an integrated circuit manufacturingdefinition dataset may cause a device as described herein to bemanufactured.

In some examples, an integrated circuit definition dataset could includesoftware which runs on hardware defined at the dataset or in combinationwith hardware defined at the dataset. In the example shown in FIG. 15 ,the IC generation system may further be configured by an integratedcircuit definition dataset to, on manufacturing an integrated circuit,load firmware onto that integrated circuit in accordance with programcode defined at the integrated circuit definition dataset or otherwiseprovide program code with the integrated circuit for use with theintegrated circuit.

The implementation of concepts set forth in this application in devices,apparatus, modules, and/or systems (as well as in methods implementedherein) may give rise to performance improvements when compared withknown implementations. The performance improvements may include one ormore of increased computational performance, reduced latency, increasedthroughput, and/or reduced power consumption. During manufacture of suchdevices, apparatus, modules, and systems (e.g. in integrated circuits)performance improvements can be traded-off against the physicalimplementation, thereby improving the method of manufacture. Forexample, a performance improvement may be traded against layout area,thereby matching the performance of a known implementation but usingless silicon. This may be done, for example, by reusing functionalblocks in a serialised fashion or sharing functional blocks betweenelements of the devices, apparatus, modules and/or systems. Conversely,concepts set forth in this application that give rise to improvements inthe physical implementation of the devices, apparatus, modules, andsystems (such as reduced silicon area) may be traded for improvedperformance. This may be done, for example, by manufacturing multipleinstances of a module within a predefined area budget.

The applicant hereby discloses in isolation each individual featuredescribed herein and any combination of two or more such features, tothe extent that such features or combinations are capable of beingcarried out based on the present specification as a whole in the lightof the common general knowledge of a person skilled in the art,irrespective of whether such features or combinations of features solveany problems disclosed herein. In view of the foregoing description itwill be evident to a person skilled in the art that variousmodifications may be made within the scope of the invention.

What is claimed is:
 1. A method of performing intersection testing of aray with a convex polygon in a ray tracing system, wherein the ray andthe convex polygon are defined in a 3D space using a space-coordinatesystem, and wherein the ray is defined with a ray origin and a raydirection, the method comprising: using a ray-coordinate system toperform intersection testing, wherein the ray-coordinate system has anorigin at the ray origin, and wherein the ray-coordinate system hasthree basis vectors, wherein a first of the basis vectors is alignedwith the ray direction; and wherein a second and a third of the basisvectors: (i) are both orthogonal to the first basis vector, (ii) are notparallel with each other, and (iii) have a zero as one component whenexpressed in the space-coordinate system; and outputting a result ofperforming the intersection testing for use by the ray tracing system.2. The method of claim 1, further comprising translating verticesdefining the convex polygon, for use in performing the intersectiontesting, by subtracting the ray origin from the positions of thevertices defining the convex polygon.
 3. The method of claim 1, whereinthe second and the third of the basis vectors of the ray-coordinatesystem have a value of ±1 as one component when expressed in thespace-coordinate system.
 4. The method of claim 1, wherein the firstbasis vector, S, when expressed with components of the space-coordinatesystem is S=A(D_(x),D_(y),D_(z)), wherein the second basis vector, P,when expressed with components of the space-coordinate system isP=B(D_(z),0,−D_(x)); and wherein the third basis vector, Q, whenexpressed with components of the space-coordinate system isQ=C(0,D_(z),−D_(y)); wherein D_(x), D_(y) and D_(z) are components ofthe ray direction in the space-coordinate system and A, B and C arescalar values.
 5. The method of claim 4, wherein B is a simplifiedfraction with D_(x) in its denominator, and C is a simplified fractionwith D_(y) in its denominator.
 6. The method of claim 3, wherein thefirst basis vector, S, when expressed with components of thespace-coordinate system is S=A(D_(x),D_(y),D_(z)), wherein the secondbasis vector, P, when expressed with components of the space-coordinatesystem is ${P = \left( {{\pm 1},0,{\mp \frac{D_{x}}{D_{z}}}} \right)};$and wherein the third basis vector, Q, when expressed with components ofthe space-coordinate system is${Q = \left( {0,{\pm 1},{\mp \frac{D_{y}}{D_{z}}}} \right)};$ whereinD_(x), D_(y) and D_(z) are components of the ray direction in thespace-coordinate system and A is a scalar value.
 7. The method of claim3, wherein the first basis vector, S, when expressed with components ofthe space-coordinate system is S=A(D_(x),D_(y),D_(z)), wherein thesecond basis vector, P, when expressed with components of thespace-coordinate system is${P = \left( {{\pm \frac{D_{z}}{D_{x}}},0,{\mp 1}} \right)};$ andwherein the third basis vector, Q, when expressed with components of thespace-coordinate system is${Q = \left( {0,{\pm \frac{D_{z}}{D_{y}}},{\mp 1}} \right)};$ whereinD_(x), D_(y) and D_(z) are components of the ray direction in thespace-coordinate system and A is a scalar value.
 8. The method of claim1, wherein the first basis vector, S, when expressed with components ofthe space-coordinate system is S=A(D_(x),D_(y),D_(z)), wherein thesecond basis vector, P, when expressed with components of thespace-coordinate system is${P = \left( {{\pm \frac{1}{D_{x}}},0,{\mp \frac{1}{D_{z}}}} \right)};$and wherein the third basis vector, Q, when expressed with components ofthe space-coordinate system is${Q = \left( {0,{\pm \frac{1}{D_{y}}},{\mp \frac{1}{D_{z}}}} \right)};$wherein D_(x), D_(y) and D_(z) are components of the ray direction inthe space-coordinate system and A is a scalar value.
 9. The method ofclaim 1, wherein the first basis vector, S, when expressed withcomponents of the space-coordinate system is S=A(D_(x),D_(y),D_(z)),wherein the second basis vector, P, when expressed with components ofthe space-coordinate system is P=(±D_(z),0,∓D_(x)); and wherein thethird basis vector, Q, when expressed with components of thespace-coordinate system is Q=(0,±D_(z),∓D_(y)); wherein D_(x), D_(y) andD_(z) are components of the ray direction in the space-coordinate systemand A is a scalar value.
 10. The method of claim 4, wherein$A = {\frac{1}{❘D_{z}❘}.}$
 11. The method of claim 1, further comprisingtransforming the ray and the convex polygon from the space-coordinatesystem into the ray-coordinate system, wherein the intersection testingis performed in the ray-coordinate system.
 12. The method of claim 1,wherein the intersection testing is performed using fixed functioncircuitry, and optionally wherein the fixed function circuitry comprisesone or more multiply-and-add components for performing multiplicationsand additions using the second and third basis vectors of theray-coordinate system.
 13. The method of claim 1, further comprisingselectively permuting the components of the ray and the verticesdefining the convex polygon, such that |D_(z)|≥|D_(x)| and|D_(z)|≥|D_(y)|, before performing intersection testing of the ray withthe convex polygon.
 14. The method of claim 1, further comprisingselectively reversing the components of the ray and the verticesdefining the convex polygon, such that D_(x)≥0, D_(y)≥0 and D_(z)≥0,before performing intersection testing of the ray with the convexpolygon.
 15. The method of claim 13, further comprising storing amapping indication of whether the mapping from the space-coordinatesystem to the ray-coordinate system affects the perceived polygonorientation, wherein the mapping indication is used to perform saidintersection testing of the ray with the convex polygon.
 16. The methodof claim 1, wherein values of at least one of: $\begin{matrix}{{\frac{D_{x}}{D_{z}}{and}\frac{D_{y}}{D_{z}}},} & (i)\end{matrix}$ $\begin{matrix}{{\frac{D_{z}}{D_{x}}{and}\frac{D_{z}}{D_{y}}},} & ({ii})\end{matrix}$ $\begin{matrix}{{\frac{1}{D_{x}}{and}\frac{1}{D_{y}}},{and}} & ({iii})\end{matrix}$ $\begin{matrix}\frac{1}{D_{z}} & ({iv})\end{matrix}$ or D_(z), are pre-computed for the ray and stored in astore, wherein the method comprises reading the stored values from thestore for use in performing intersection testing.
 17. The method ofclaim 1, wherein the outputted result is used in the ray tracing systemfor rendering an image of a 3D scene.
 18. A method of performingintersection testing of a ray with a convex polygon in a ray tracingsystem, wherein the ray and the convex polygon are defined in ann-dimensional space using a space-coordinate system, wherein n≥3, andwherein the ray is defined with a ray origin and a ray direction, themethod comprising: using a ray-coordinate system to perform intersectiontesting, wherein the ray-coordinate system has an origin at the rayorigin, and wherein the ray-coordinate system has n basis vectors,wherein a first of the basis vectors is aligned with the ray direction,wherein (n−1) of the basis vectors are orthogonal to the first basisvector, wherein no pairing of the (n−1) basis vectors are parallel witheach other, and wherein the (n−1) basis vectors have zeros for (n−2)components when expressed in the space-coordinate system; and outputtinga result of performing the intersection testing for use by the raytracing system.
 19. An intersection testing module, for use in a raytracing system, configured to perform intersection testing of a ray witha convex polygon, wherein the ray and the convex polygon are defined ina 3D space using a space-coordinate system, and wherein the ray isdefined with a ray origin and a ray direction, the intersection testingmodule being configured to: use a ray-coordinate system to performintersection testing, wherein the ray-coordinate system has an origin atthe ray origin, and wherein the ray-coordinate system has three basisvectors, wherein a first of the basis vectors is aligned with the raydirection; and wherein a second and a third of the basis vectors: (i)are both orthogonal to the first basis vector, (ii) are not parallelwith each other, and (iii) have a zero as one component when expressedin the space-coordinate system; and output a result of performing theintersection testing for use by the ray tracing system.
 20. Theintersection testing module of claim 19, wherein the first basis vector,S, when expressed with components of the space-coordinate system isS=A(D_(x),D_(y),D_(z)), wherein the second basis vector, P, whenexpressed with components of the space-coordinate system isP=B(D_(z),0,−D_(x)); and wherein the third basis vector, Q, whenexpressed with components of the space-coordinate system isQ=C(0,D_(z),−D_(y)); wherein D_(x), D_(y) and D_(z) are components ofthe ray direction in the space-coordinate system, A is a scalar value,and either: $\begin{matrix}{{B = {{{\pm \frac{1}{D_{x}}}{and}C} = {\pm \frac{1}{D_{y}}}}},{or}} & (i)\end{matrix}$ $\begin{matrix}{B = {{{\pm \frac{1}{D_{z}D_{x}}}{and}C} = {\pm {\frac{1}{D_{z}D_{y}}.}}}} & ({ii})\end{matrix}$